Download Forensic Interpretation of Glass Evidence

Forensic
Interpretation
of Glass
Evidence
©2000 CRC Press LLC
Forensic
Interpretation
of Glass
Evidence
James Michael Curran, Ph.D.
Tacha Natalie Hicks, Ph.D.
John S. Buckleton, Ph.D.
with contributions by
José R. Almirall • Ian W. Evett • James A. Lambert
CRC Press
Boca Raton London New York Washington, D.C.
©2000 CRC Press LLC
Library of Congress Cataloging-in-Publication Data
Curran, James Michael.
Forensic interpretation of glass evidence / by James Michael Curran, Tacha Natalie
Hicks, John S. Buckleton.
p. cm.
Includes bibliographical references and index.
ISBN 0-8493-0069-X (alk.)
1. Glass. 2. Forensic engineering. 3. Ceramic materials. 4. Bayesian statistical decision
theory. I. Hicks, Tacha Natalie. II. Buckleton, John S. III. Title.
TA450.C87 2000
616′.1—dc21
00-030354
CIP
This book contains information obtained from authentic and highly regarded sources. Reprinted material
is quoted with permission, and sources are indicated. A wide variety of references are listed. Reasonable
efforts have been made to publish reliable data and information, but the author and the publisher cannot
assume responsibility for the validity of all materials or for the consequences of their use.
Neither this book nor any part may be reproduced or transmitted in any form or by any means, electronic
or mechanical, including photocopying, microfilming, and recording, or by any information storage or
retrieval system, without prior permission in writing from the publisher.
The consent of CRC Press LLC does not extend to copying for general distribution, for promotion, for
creating new works, or for resale. Specific permission must be obtained in writing from CRC Press LLC
for such copying.
Direct all inquiries to CRC Press LLC, 2000 N.W. Corporate Blvd., Boca Raton, Florida 33431.
Trademark Notice: Product or corporate names may be trademarks or registered trademarks, and are
used only for identification and explanation, without intent to infringe.
© 2000 by CRC Press LLC
No claim to original U.S. Government works
International Standard Book Number 0-8493-0069-X
Library of Congress Card Number 00-030354
Printed in the United States of America 1 2 3 4 5 6 7 8 9 0
Printed on acid-free paper
©2000 CRC Press LLC
Preface
In 1933 the problem of identifying certain minute splinters of glass was
referred to the Dominion Laboratory in New Zealand. The splinters from
the corner of an attaché case of the accused were alleged to be the result of
the case being used to break a shop window preparatory to taking goods.1
Refractive index and density were used in the comparison. Of 65 samples
previously encountered only 1 matched in all respects to the glass from the
attaché case and that 1 was the plate glass from the shop window.
The analysis of glass evidence for forensic uses was an exciting topic
even before Nelson and Revell2 first carried out their backward fragmentation experiments and Ojena and De Forest3 perfected the method of characterizing a fragment of glass by its refractive index.
This field of glass evidence interpretation was revolutionized in the late
1970s when a young Forensic Science Service (FSS) document examiner
named Ian Evett4 decided to introduce statistics as a method for consistent
and objective evaluation of forensic glass evidence.
Evett discussed the problems facing forensic scientists with Professor
Dennis Lindley. Lindley can be considered as one of the forefathers of modern Bayesian thinking, given that he regularly entered into debate over the
validity of the subject with the great Sir Ronald Aylmer Fisher who is considered the founder of modern statistics. At the same time that Dr. Evett
introduced statistics into the analysis of glass evidence, Professor Lindley
remarked that the actual solution was a Bayesian one.5
This book evolved from an “interpretation manual” written by Evett,
Lambert, and Buckleton for the FSS. Some material has been added and other
areas updated. We also acknowledge Dr. José Almirall for the substantial
amount of material he contributed to Chapter 1.
We have followed the Bayesian method of reasoning in this book. This
has been a deliberate choice, and we have not argued its advantages extensively here. It may become apparent to anyone reading this book what these
advantages are simply by noticing that the Bayesian approach has allowed
us to handle some difficult casework problems with the assistance of logic.
It would complete the picture to give a critique of the frequentist approach.
This is surprisingly hard because the frequentist approach is not firmly
grounded in logic. No person has offered any coherent and comprehensive
system by which the frequentist approach may be applied to highly variable
©2000 CRC Press LLC
D.F. Nelson and B.C. Revell, Backward fragmentation from breaking glass, J.
Forensic Sci. Soc., 7, 58, 1967 (Reprinted with the kind permission of the Forensic
Science Society).
casework situations, or how this frequency applies to the questions before
the court.
By using either method, most practitioners can interpret simple evidence
such as one bloodstain left at a scene that matches a suspect. However,
misleading statements and ad hoc solutions may result when the frequentist
approach is applied to more complex cases. For instance, we have shown
coherent methods to understand the value of the presence of glass per se.
We have seen frequentists attempt to formulate this assessment, and both
they and we feel that there is great value in this. Furthermore, both the
Bayesian and frequentist schools would accept that the larger the group of
©2000 CRC Press LLC
glass, the higher the value of the evidence (in most cases). The difference is
that the Bayesian school can make a logical attempt to evaluate this evidence.
It is possible to expose these differences with a simple case example.
Consider a case where a man has been seen to shoulder charge six windows.
A suspect is apprehended 30 minutes later and found to have two fragments
of glass upon his highly retentive black jersey. These two fragments match
one control. Simply quoting the frequency with which this match would
occur, say 1%, might imply some evidence supporting the prosecution hypothesis. Presented with this case, frequentists perform one of several actions. The more thoughtful of them start to say something along the lines
that this does not appear to be much glass given the circumstance. In doing
this, they are unconsciously evolving toward the Bayesian thought.
When computing power and the statistical tools required became available in the early 1980s, Ian Evett, with the help of John Buckleton, Jim
Lambert, Colin Aitken, and Richard Pinchin, developed an approximate
Bayesian solution which was implemented and used in forensic casework.6,7
We call this implementation approximate for two reasons: (1) a full Bayesian
treatment requires the evaluation of the entire joint distribution of the control
and recovered samples (a task which still defies solution today), and (2) the
approach still contains a “match”/“nonmatch” step. This second point was
remedied by Walsh et al.8 in the mid 1990s.
Since then there have been rapid advances in the statistical analysis of
forensic glass evidence and the statistical evaluation of many other types of
forensic evidence, the most notable of these being DNA. In the U.S. and
many other countries around the world, forensic scientists have found the
major focus of their work shifting toward the evaluation of DNA evidence.
This initially involved the analysis of nuclear DNA found in blood, semen,
and saliva, but in recent years mitochondrial DNA has also become an
important source of forensic information. However, law enforcement is beginning to realize that DNA is not always available and there is a whole
field of forensic science called trace evidence analysis.
It is for this reason that the authors decided to write a book on the
statistical interpretation of glass evidence. Glass work accounts for some 20%
of casework in New Zealand and 12% in the U.K. In the past, the U.S. has
not had a strong history of forensic glass analysis, but with large amounts
of federal funds formerly devoted to military applications being set aside
for forensic research, we expect this to change rapidly over the next few
years. The U.S. is at the forefront of research in the use of elemental concentration data as a means of discriminating between different glass sources.
This book is intended for forensic scientists and students of forensic
science. There have been enough treatises written by statisticians for statisticians. Of the six contributors, five are forensic scientists, one is a statistician,
and all have done physical glass analysis and presentations of statistical
evidence in a court of law. The intention of this book is to provide the
practicing forensic scientist with the necessary statistical tools and methodology to introduce statistical analysis of forensic glass evidence into his or
©2000 CRC Press LLC
her lab. To that end, in conjunction with this book, we offer free software
(available by E-mailing James Curran at: [email protected],
courtesy of James Curran and ESR) that implements nearly all of the methodology in this book. We offer this software with the minor proviso that
users must supply their own data because we are unable to give away some
of the data sets mentioned in this book.
The examples and theory in this book primarily revolve around refractive index measurements. However, where applicable, the methods have
been extended for elemental concentration data. Caseworkers who deal with
mostly elemental analysis should not be discouraged by this apparent slant.
The theory is easily transferred to the elemental perspective in most cases
by simply substituting the equivalent elemental measurement.
Chapter 1 is an introduction to the physical properties and methods for
analysis of forensic glass analysis. This chapter is intended for forensic scientists new to the area of glass analysis, students of forensic science, and
perhaps statisticians or lawyers who are interested in the physical processes
behind the data.
Chapter 2 provides an introduction and review of the conventional or
classical approaches to the statistical treatment of forensic glass evidence.
Topics covered include range tests, hypothesis tests, and confidence intervals. Grouping of glass considered to have come from multiple sources, ttesting, coincidence probabilities, and a simple extension of the classical
approach to the analysis of elemental data. By the end of Chapter 2, the
reader should be able to perform a two-sample t-test for the difference
between two means, carry out a range test, construct a confidence interval,
and have a basic understanding of the statistical procedures necessary for
automatic grouping of recovered glass samples.
Chapter 3 offers the reader an introduction to the application of Bayesian
statistics to forensic science. Bayesianism is an entirely different approach to
the subject of statistics. A Bayesian approach often requires more thought
about the problem involved and the results desired. In this chapter the reader
will learn the reasoning behind the Bayesian methods and hopefully gain
some insight as to why this approach is preferred. In addition to Bayesian
thinking, Chapter 3 introduces the rules of probability, the details of a Bayesian approach to the statistical analysis of forensic glass evidence, and many
worked examples to aid comprehension.
In Chapter 4 we attempt to summarize the experimental knowledge
gained to date in the fields of glass frequency surveys and the prevalence
of glass on clothing.
Chapter 5 describes survey work on transfer and persistence of glass.
Many of these works will be familiar to the experienced glass examiner;
however, they are presented with the deliberate attempt to make them relevant in the Bayesian framework and with some novel comparative work.
Chapter 6 discusses the particular statistical tools and data that are
necessary for a Bayesian approach to be implemented. In particular, the
©2000 CRC Press LLC
reader is introduced to histograms and their more robust extension and
density estimates. This chapter also discusses the various software packages
available for the evaluation of glass evidence.
Chapter 7 covers the difficult task of reporting statistical information in
statements and viva voce evidence. It covers the “fallacy of the transposed
conditional,” an error that has led to appeals in the field of DNA evidence.
©2000 CRC Press LLC
The Authors
James M. Curran, Ph.D., is a statistics lecturer at the University of Waikato,
Hamilton, New Zealand, with extensive publications in the statistical analysis of forensic evidence. In 1994 he was awarded a scholarship from the
Institute of Environmental Science Ltd. (ESR) to work on statistical problems
in the analysis of forensic glass evidence. He received his Ph.D. from the
University of Auckland, New Zealand, in 1997. During this period, Dr.
Curran lectured and gave seminars at numerous conferences and universities
in New Zealand and overseas. He also developed software that is now used
for day-to-day casework in New Zealand and as a research tool in several
laboratories around the world.
In 1997 Dr. Curran was awarded a postdoctoral fellowship from a New
Zealand government agency, the Foundation for Research in Science and
Technology (FORST). This provided Dr. Curran with funding to go to North
Carolina State University in Raleigh, NC, for 2 years and work on statistical
problems in DNA evidence. While in the U.S., Dr. Curran provided statistical
reports or expert testimony in nearly 20 criminal cases involving DNA evidence.
The California Association of Criminalists and the U.K. Forensic Science
Service awarded Dr. Curran the Joint Presidents Award for significant contribution to the field of forensic science by a young practitioner.
Tacha Hicks, Ph.D., graduated with honors in forensic science from the
Institut de Police Scientifique et de Criminologie (IPSC) at the University of
Lausanne. Her doctoral dissertation entitled “The Interpretation of Glass
Evidence” explores many areas described in this book such as transfer and
persistence, glass found at random, and knowledge-based systems.
Since 1996 Dr. Hicks has been involved in both the European Glass
Group (EGG) and the Scientific Working Group on Materials (SWGMAT).
From 1993 to 1999 Dr. Hicks worked as a research assistant at the IPSC,
supervising student research in glass evidence.
©2000 CRC Press LLC
John S. Buckleton, Ph.D., is a forensic scientist working in New Zealand.
He received his Ph.D. in chemistry from the University of Auckland, New
Zealand. Dr. Buckleton has coauthored more than 60 publications in the field
of forensic science and has taught this subject internationally. Dr. Buckleton
has helped develop a computerized expert system for the interpretation of
glass evidence.
In 1992 Dr. Buckleton was awarded the P.W. Allen Award for the best
paper in the Journal of the Forensic Science Society.
©2000 CRC Press LLC
Contents
Chapter
1.1
1.2
1.3
1.4
1.5
1.6
1.7
1.8
1.9
1.10
1.11
1 Examination of Glass
History
Flat Glass
Float Glass
Toughened Glass
Laminated Glass
Glass Composition
Glass Breakage Under Impact
1.7.1 Breakage in Flexion
1.7.2 Determination of Side of Impact
1.7.3 Percussion Cone Breakage
1.7.4 Transfer of Glass
Physical Examinations
Examinations of Large Fragments
1.9.1 The Comparison of Thickness
1.9.2 The Comparison of Color
1.9.3 Matching Edges and Matching Hackle Marks
1.9.4 Density Comparisons
Examinations Performed on Small and Large Fragments
1.10.1 Recovering Glass
1.10.2 Examination of Transparent Material to Determine
Whether It Is Glass
1.10.3 The Examination of Surface Fragments
1.10.4 Refractive Index Determinations
1.10.5 Dispersion
1.10.6 Refractive Index Anomalies
1.10.7 The Examination of Tempered (Toughened) Glass
by Annealing
Elemental Composition
1.11.1 X-Ray Methods
1.11.1.1 Classification of Glass Using
X-Ray Methods
1.11.1.2 Discrimination of Glass Using
X-Ray Methods
©2000 CRC Press LLC
1.11.2 ICP Techniques
1.11.2.1 Classification of Glass Using
ICP Techniques
1.11.2.2 Discrimination of Glass Using
ICP Techniques
1.12 Summary
1.13 Appendix A — Snell’s Law
Chapter 2 The Conventional Approach to Evidence Interpretation
2.1 Data Comparison
2.1.1 Range Tests and Use of Confidence Intervals
2.1.2 Confidence Interval
2.2 Statistical Tests and Grouping
2.2.1 Grouping
2.2.1.1 Agglomerative Methods
2.2.1.2 Divisive Methods
2.2.1.3 Performance
2.2.2 Statistical Tests
2.2.2.1 Hypothesis Testing
2.2.2.1.1 Student’s t-Test
2.2.2.1.2 Welch’s Modification to the
Student’s t-Test
2.2.2.2 How Many Control Fragments?
2.2.2.3 Setting Significance Levels
2.2.2.4 Elemental Composition Measurements —
Hotelling’s T2
2.2.2.4.1 The Multiple Comparison
Problem
2.2.2.4.2 Hotelling’s T2 — A Method for
Comparing Two Multivariate
Mean Vectors
2.2.2.4.3 Examples
2.2.2.4.4 Discussion on the Use of
Hotelling’s T2
2.3 Coincidence Probabilities
2.4 Summary
2.5 Appendix A
2.6 Appendix B
2.7 Appendix C
Chapter 3 The Bayesian Approach to Evidence Interpretation
3.1 Probability — Some Definitions
3.2 The Laws of Probability
3.2.1 The First Law of Probability
©2000 CRC Press LLC
3.3
3.4
3.5
3.6
3.7
3.8
Chapter
4.1
4.2
4.3
4.4
4.5
4.6
4.7
4.8
4.9
4.10
3.2.2 The Second Law of Probability
3.2.3 The Third Law of Probability
3.2.4 The Law of Total Probability
3.2.5 Bayes Theorem
3.2.6 The Relationship Between Probability and Odds
3.2.7 The Odds Form of Bayes Theorem
Bayesian Thinking in Forensic Glass Analysis
3.3.1 A Generalized Bayesian Formula
Taking Account of Further Analyses
Search Strategy
Comparison of Measurements: The Continuous Approach
3.6.1 A Continuous LR Approach to the Interpretation of
Elemental Composition Measurements from Forensic
Glass Evidence
3.6.1.1 The Continuous Likelihood Ratio for
Elemental Observations
3.6.1.2 Examples
3.6.1.3 Discussion
Summary
Appendix A
4 Glass Found at Random and Frequency of Glass
Relevant Questions
Availability
Glass Found at Random (Clothing Surveys)
4.3.1 Glass Found on the General Population
4.3.1.1 Glass Recovered on Garments
4.3.1.2 Glass Recovered on Shoes
4.3.1.3 Glass Recovered in Hair
4.3.2 Glass Recovered on the Suspect Population
4.3.2.1 General Trends
4.3.2.2 Glass Recovered on Shoes
Comparison Between Suspect and General Populations:
An Example
Estimation of the Probability of Finding at Random i Groups
of j Fragments
Frequency of the Analyzed Characteristics
Control Glass Data Collections
Clothing Surveys
Characteristics of Glass Found on the General Population
4.9.1 Glass Recovered on Garments
4.9.2 Glass Recovered on Shoes
Characteristics of Glass Found on the Suspect Population
©2000 CRC Press LLC
4.11 Comparison Between Suspect and General Populations:
An Example
4.12 Summary
Chapter 5 Transfer and Persistence Studies
5.1 Transfer of Glass
5.1.1 Transfer of Glass to the Ground
5.1.1.1 Number, Size, and Distribution of the
Fragments
5.1.1.2 Influence of the Window Type and Size
5.1.1.3 Presence of an Original Surface
5.1.2 Transfer of Glass Broken with a Firearm
5.1.3 Transfer of Vehicle Glass
5.1.4 Transfer of Glass to Individuals Standing Nearby
5.1.5 Transfer of Window Glass to Garments
5.1.6 Transfer of Glass with a Pendulum
5.1.7 Glass Broken Under Conditions Similar
to Casework
5.1.8 Transfer of Vehicle Glass and Absence of Glass
5.1.9 Transfer of Glass When a Person Enters Through a
Window
5.1.10 Influence of the Weather on Transfer
5.1.11 Transfer of Broken Glass
5.1.12 Transfer of Window Glass to Hair
5.1.13 Transfer of Window Glass to Footwear
5.1.14 Secondary and Tertiary Transfer
5.1.15 Transfer: What Do We Know?
5.2 Persistence of Glass on Garments
5.2.1 Early Studies
5.2.2 Persistence of Glass on Clothing
5.2.3 Persistence of Glass on Shoes
5.2.4 Persistence of Glass in Hair
5.3 Main Results of the Studies
5.4 Modeling Glass Transfer and Making Estimates
5.4.1 Graphical Models
5.4.2 A Graphical Model for Assessing Transfer
Probabilities
5.4.3 Results
5.4.4 Conclusions from the Modeling Experiment
5.5 Appendix A — The Full Graphical Model for
Assessing Transfer Probabilities
5.6 Appendix B — Probabilistic Modeling and Quantitative
Assessment
©2000 CRC Press LLC
Chapter 6 Statistical Tools and Software
6.1 Data Analysis
6.1.1 Histograms and Lookup Tables
6.1.1.1 Constructing a Histogram
6.1.2 Constructing a Floating Window
6.1.3 Estimating Low Frequencies
6.1.4 Density Estimates
6.1.4.1 Random Variables and Probability Density
Functions
6.1.5 Kernel Density Estimators
6.1.5.1 What Is a Good Tuning Parameter for a
Kernel Density Estimator?
6.2 Calculating Densities by Hand
6.3 Computer Programs
6.3.1 The Fragment Data System (FDS)
6.3.2 STAG
6.3.3 Elementary
6.3.4 CAGE
6.4 Summary
6.5 Appendix A
Chapter 7 Reporting Glass Evidence
7.1 Verbalization of a Likelihood Ratio Answer
7.2 Sensitivity of the Likelihood Ratio Answer to Some
of the Data Estimates
7.3 The Effect of Search Procedures
7.4 Fallacy of the Transposed Conditional
References
©2000 CRC Press LLC
chapter one
Examination of glass
José Almirall, John Buckleton, James Curran, and Tacha Hicks
A forensic scientist may be asked to examine glass to reconstruct events (for
example, to determine whether a pane of glass was broken from the inside
or outside) or to associate a person or an object with the scene of a crime or
a victim. The strength of the opinion of the forensic scientist depends on
many factors, including the nature of the background information and comparisons used, the type of glass involved, whether it is rare or common, and
how well it is characterized.
This chapter will consider the examinations and comparisons that are
often used in the evaluation of glass as evidence. This is preparatory to
introducing modern and classical methods for the interpretation of the allegedly transferred material. We also include a brief description of glass manufacturing processes and composition of glass, as these are important in
understanding the examinations to be performed and the sources of variation
to be expected. We cover measurements such as comparison of thickness,
color, density, and more extensively the measurement of refractive index and
elemental analysis, two of the dominant techniques in modern laboratories.
It is our intention, however, to introduce these methods rather than give an
extensive review of them.
1.1 History
Glass is defined as “an inorganic product of fusion that has cooled to a rigid
condition without crystallization.”
The first glass objects used by man probably originated from naturally
occurring sources such as obsidian, from which sharp tools may be chipped.
It is believed that manmade glass originated in the regions now known as
Egypt and Iraq around 3500 years ago.9 Pliny, the Roman historian, recounts
a story whereby Phoenician sailors propped a cooking pot on some blocks
©1999 CRC Press LLC
©2000 CRC Press LLC
of natron (an alkali). They noticed that the sand beneath the fire had melted
and assumed the properties of a liquid. Upon cooling, they also noticed that
the liquid hardened into the material now known as glass. Modern scholarship suggests that glass developed from faience, an older material made
from crushed quartz and alkali. These same ingredients in different proportions form a true glass.
1.2 Flat glass
The development of flat glass manufacturing methods progressed through
the early 20th century, first in Belgium with the Fourcault process.10 In this
process, the components are first melted in a large tank (which little by little
dissolves itself in the molten glass, so that traces of the tank [zirconium, for
example] are present in the glass product). The sheet of glass is then drawn
vertically through a slotted refractory in a continuous ribbon. The problem
of “necking down” to a narrow ribbon was solved by chilling the ribbon
against paired rolls that gripped the sheet edges. Glass made this way has
a fire finish or polish which is the brilliant surface achieved by allowing
molten glass to cool to rigidity without contact. The solidified ribbon, however, has a noticeable amount of waviness. Figure 1.1 shows a diagrammatic
sketch of the Fourcault process.
Drawing machine
and vertical lehr
Sheet coolers
Depressed
Debiteuse
Glass
Figure 1.1
Diagrammatic sketch of the Fourcault process.
©2000 CRC Press LLC
Later in North America the Libbey-Owens and Pittsburgh methods
refined this. The latter replaces the depressed debiteuse with a submerged
drawbar, but otherwise the principle and outcome are the same.
Glass drawn from a molten source in this manner is referred to as sheet
glass and typically has a fine finish with optical distortions due to minor
thickness variations. The term “sheet glass” has been used in forensic texts
to mean either any flat glass or glass made by this wire drawn method. This
latter definition seems better and will be used in this book.
Sheet glass has a minor amount of visual distortion due to minor variations in thickness. Plate glass was developed to overcome the problem of
visual distortion. Typical plate glass production has two predominant steps:
first, rough blank production and, second, the grinding and polishing stage.
Plate glass was possibly first produced by pouring onto a flat base and rolling
to produce the rough blank. A factory in St. Gobain, France was built for
this purpose in 1688. Modern machines produce the rough blank by rolling
glass directly from the spout and automate grinding and polishing.
Patterned glass may be produced by a number of methods, but mass
produced patterned glass is typically made by squeezing the flow of molten
glass between paired water-cooled rollers. One or both rollers impress the
desired pattern. Wired glass can also be made by feeding wire mesh between
the forming rollers.
1.3 Float glass
The float process was developed by the Pilkington company in Britain. The
incorporation of a liquid tin bed in the float process leads to a smooth and
flat surface. The molten glass is delivered onto a bed of liquid tin where the
glass “floats” over the metal. Rollers on the sides pull the glass to a desired
thickness depending on the speed of the pull. The float process is currently
used for the manufacture of the vast majority of flat glass (see Figure 1.2).
It is important to note that the surface in contact with the tin will show
luminescence when excited at 254 nm, enabling the forensic scientist to
identify float glass (elemental analysis with the scanning electron microscope
[SEM] also allows the identification of float glass). It has also been shown
that this manufacturing process induces a gradient in the refractive index of
the glass, which becomes anisotropic.
1.4 Toughened glass
Flat glass may be processed by tempering. This method is also termed
“toughening” and indeed does produce a product that is harder to break.
The product is sometimes called safety glass, which reflects the way that it
breaks into “diced” pieces that do not have sharp or pointed edges. Tempering is achieved by rapid cooling of the heated glass, typically with air
©2000 CRC Press LLC
Figure 1.2
1980
1979
1978
1977
1976
1975
1974
1973
1972
1971
1970
1969
1968
1967
1966
1965
1964
Millions of ft2
4000
3500
3000
2500
2000
1500
1000
500
0
Flat glass production in the U.S. from 1964 to 1980.
jets on both sides. This compresses the surfaces and places the center in
tension. The process is believed to strengthen (toughen) the glass because
surface flaws are thought to initiate breakage. Flaws initiate breakage when
the glass is in tension. Inducing surface compression reduces or eliminates
this tendency. The center, in tension, cannot possess surface flaws. Modern
trends in tempering include the move to thinner glass (a cost-saving measure) and quick sag bending, a shaping technique.
Large fragments of tempered glass may appear in casework in hit and
run incidents or more rarely in the pockets of clothing. Such large fragments
may be recognized as tempered by their diced appearance and by the presence of a frosted line through the middle of the cube (see Figure 1.3).
Figure 1.3 Characteristics of tempered glass cubes, such as frosting in the middle
of the cube.
The identification of glass as “toughened” by polarized light microscopy
has been reported for fragments measuring more than 20 mm3.11 Sanger
concluded that correctly mounted and lit fragments exhibiting brightness, if
not colors, must be toughened. Small fragments not exhibiting brightness
may still be toughened, but below the size required to produce the effect.
The authors are unaware of any laboratory using this method. With the small
©2000 CRC Press LLC
fragments typical of glass recovered from clothing, these observations cannot
be made, and for fragments of this size annealing has become the method
of choice.
1.5 Laminated glass
Another modern trend is the increased use of laminated glass. This is a
composite typically of a sheet of plastic between two sheets of flat glass. Flat
or shaped laminates may be produced and are dominant in modern windscreens. The two sheets may be taken from one pallet, in which case they
are typically very similar, or from two pallets, in which case they may differ.
Some applications deliberately call for color differences in the two sheets.
Laminates are noted for their ability to resist penetration, acting as security
glass or to restrain passengers within a vehicle during collision.
1.6 Glass composition
Soda-lime glass accounts for the majority of the glass manufactured around
the world, producing windows (flat glass) for buildings and automobiles
and containers of all types. The major components in the raw materials for
this type of glass are sand (SiO2, 63 to 74%), soda ash (Na2CO3, 12 to 16%),
limestone (CaO, 7 to 14%), and cullet. The major component (or primary
former) is sand, and if it is of very high quality then it is sufficient to produce
glass. However, because of the high purity and temperature required to melt
sand, soda ash and potassium oxides are added to lower the melting point.
Limestone is then used so that the glass becomes insoluble in water and
acquires a higher chemical durability. As the products used are not pure,
other processes designed to improve purity may be necessary. Of particular
concern for the manufacturer is iron or chromium contamination in the sand
deposits, which can lead to undesirable coloring.* The impurities in the main
products, such as those in cullet, the potassium oxides in soda ash, the
magnesium as well as aluminum oxides in the lime, or strontium in dolomite
(a source of CaO and MgO), can be used by the forensic scientist to discriminate glass of the same type.
Depending on the end use of the product and on the manufacturing
process, special components will be intentionally used.
• Boron oxide (B2O3), for example, will be added to improve heat durability in glass intended for cookware, laboratory glassware, or automobile headlamps.
• Lead oxide (PbO) will improve the sparkling effect of glass or absorb
radiation (it is also used for stemware, neon, electrical connections,
thermometers, and eyeglasses).
* The manufacturer will add CeO2 (As2O3 was formerly used), SbO, NaNO3, BaNO3, K2SO4, or
BaSO4 to decolorize the glass.
©2000 CRC Press LLC
• Other examples are the addition of silver (Ag) in sunglasses and of
strontium in television screens in order to absorb radiation.
Depending on its use, glass will also contain different quantities of the
same elements. For example, container glass will have a low level of magnesium, iron, and sodium compared to flat glass.
1.7 Glass breakage under impact
In order to understand transfer mechanism, we will discuss breakage under
impact. There is a separate body of knowledge relating to breakage under
heat stress. As a preliminary rule, glass is strong in compression, but weak
in tension. This fact in itself explains a lot regarding glass breakage. However,
there are two different ways in which glass may fail under impact.12 Failure
to recognize these two different mechanisms has led to some erroneous
thinking. For instance, a toughened “glass will arrest the fall from a couple
of stories height of a massive steel ball…but it will collapse like a bubble
from the impact of a little steel dart dropped from the height of four inches.”
This example defines the two fracture mechanisms: the flexural type or the
bump check (percussion cone) type.
Despite their names, both types can be produced under static conditions
or under impact. Glass breaking under flexion will typically have a break
origin opposite the point of stress (or impact) (see Figure 1.4).
Origin of breakage under impact
Figure 1.4
The origin of breakage.
The effect of “flaws” as points of initiation has long been known.13 Flaws,
especially on the surface opposite to the force, greatly reduce the strength
of the glass to flexion breakage.
Flexural stresses are essentially the same for any object and are determined more by the kinetic energy of the striking object rather than by the
nature of the object itself. Therefore, the stress imparted by a wooden bat
and a steel ball is similar with respect to flexural stress. Bearing stress and
©2000 CRC Press LLC
the tendency to produce percussion cones, however, is very different for
these two objects.
Key elements, therefore, in examining breakage are flexibility vs.
strength of the glass and the nature of the impacting object. An object cannot
start a percussion flaw if it is too soft to do so.
Small, hard objects moving rapidly do damage mainly by starting percussion cones (however, refer to the discussion of rifle or gun shots later),
whereas massive objects (of any hardness) traveling slowly initiate breakage
in flexion.
1.7.1 Breakage in flexion
The concept of breakage in flexion is essentially the same in flat glass and
container glass except that in container glass the principle is modified by
the presence of flexibility restricting parts of the bottle such as the base of
the bottle. Here we consider flat glass.
Remember that glass is weak in tension and strong in compression.
Consider then a pane of glass under flexion pressure held firmly at the edges
(see Figure 1.5). Point A is in tension and is likely to be the origin of the
breakage. The opposite side is in compression and will typically not initiate
breakage (under flexion stress). Radial cracks will begin at point A and will
radiate out in several directions. Points B and C are also in tension and will
typically initiate concentric fractures. Hence, the “spider’s web” appearance
of flat glass after flexion breakage.
B
C
A
Figure 1.5
The origin of breakage under flexion.
1.7.2 Determination of side of impact
This determination is performed by an examination of ridges on the broken
surfaces of the glass. These ridges are termed rib or hackle marks. According
to Preston,14 the long curved lines have been termed rib marks, while the
shorter straight lines are hackle marks (see Figure 1.6).
©2000 CRC Press LLC
Figure 1.6
Hackle mark comparison for individualization of fragments.
The fracture always approaches the rib mark from the concave side and
leaves from the convex side. In a region of tension the fracture will travel
rapidly, leaving the rib marks well spaced, whereas in an area of compression
they will be closer together.
A single, well-defined rib occurs at the termination of the fracture and
is a clue as to whether a fracture has extended subsequently.
To perform this examination, the glass remaining in the frame should
be labeled inside and outside and much of the glass from the ground should
be retained. The examiner begins by reconstructing the pane until the point
of impact is obvious. The examiner should not rely on the triangular shape
of a single piece or of cratering to determine the point of origin, but rather
should reconstruct the pattern of radial and concentric cracks. Then the “four
Rs” rule may be applied.15,16
Ridges on Radial cracks are at Right angles to the Rear.
The origin of the rule can be seen by considering the tension and compression sides of the glass. The closely spaced rib marks are on the compression side, whereas widely spaced ones are on the tension side. Typically, the
tension side is at the rear in flexion breakage for the radial cracks; therefore,
the widely spaced ridges are to the rear.
This rule is unreliable on laminated glass. However, the side of impact
is usually obvious in laminated glass due to the fact that it remains deformed
after breakage.
©2000 CRC Press LLC
Fractures in tempered glass appear to originate in the central region
regardless of where the point of impact was. The surfaces of tempered glass
are in compression, whereas the central region is in tension. This makes
interpretation of hackle marks in tempered glass difficult.17
Photography of rib or hackle marks is facilitated by shadow photography.17
1.7.3 Percussion cone breakage
Perhaps the most obvious small, high velocity, and dense object that may
strike glass is a bullet (see Figure 1.7), although cases involving sling shots
and other projectiles are known.
Figure 1.7
Cratering effects on the exit side of a bullet shot through glass.
Experiments on shooting through glass show fractures both of the flexion
type and the percussion cone type. “Flexure breaks, essentially radial cracks,
form first and then the percussion cone is driven through to the rear side.
This is attested by the fact that the conical flaw was not smooth and perfect,
but made up of radial segments which terminated against the radial
cracks.”12
Care should be taken if the projectile has not penetrated the glass, as the
larger side of the percussion cone may be toward the point of impact.
When two projectiles penetrate a pane of glass the sequence can be
determined from the observation that the cracks for the second impact terminate at those caused by the first. Care should be taken as some cracks may
“extend” by any movement after the incident.
©2000 CRC Press LLC
1.7.4 Transfer of glass
In 1967 Nelson and Revell2 reported the backward fragmentation of glass
and demonstrated it by photography. Backward fragmentation is the scatter
toward the direction of the force and has since become known as backscatter.
They broke 19 sheets of glass and observed backward fragmentation in all
cases. They observed that “whenever a window is broken, it is to be expected
that numerous fragments of glass will strike any person within a few feet
of the window.”
The size of the fragments that are likely to be transferred from crime
scenes depends on the type of glass that was broken and many other factors
that are discussed later. However, there are two types of crime that have a
reasonable chance of producing large fragments for comparison. These are
hit-and-run incidents (incidents where a vehicle has collided with another
vehicle, pedestrian, or other object) and ram raids (incidents where a vehicle
has been used to effect entry). In hit-and-run cases, broken tempered glass
produces relatively large pieces often scattered at (or about) the point of
impact. These pieces usually include original surfaces; full thickness measurements are also possible.
Ram raids may result in large pieces of window glass being transferred
to a vehicle.
However, in the bulk of cases of glass breakage, glass transfer to clothing
involves small fragments with their size ranging from 0.1 to 2 mm and their
shapes being typically irregular.
1.8 Physical examinations
One of the determining factors in the choice of analytical scheme employed
in any given case is the size of recovered fragments. If the fragments are
large, then many possibilities exist that are not realistic for the smaller fragments typically recovered from clothing. We will first present the physical
examinations that can be performed on large fragments.
1.9 Examinations of large fragments
1.9.1 The comparison of thickness
Occasionally, fragments of glass are recovered that exhibit the full thickness
of the source object. These can be compared by direct thickness measurement. Once measured, it may be necessary to compare two sets of measurements. Unlike refractive index, no body of statistical work exists in this area,
although similarities in the comparison problem suggest that a similar solution is plausible.
It would be tempting to compare two sets of measurements directly by
determining whether the ranges overlap or by use of a t-test or similar
statistical test. However, sampling issues are quite serious with respect to
©2000 CRC Press LLC
thickness. Take for instance a toughened windscreen thought to have been
the source of glass on a road at the point of impact with a pedestrian. The
glass samples from the car are likely to be from the edges of the windscreen
and those on the road from the center. Given the way in which windscreens
are made and shaped, it is plausible that the edges are a different thickness,
albeit slightly, to the center due to the bending in the production of the
windscreen.
In this section we shall restrict ourselves to a subjective approach along
the lines of “how close must two sets of thickness measurement be for them
to be from the same source.”
Renshaw and Clarke18 give the standard deviation they encountered in
seven vehicle windscreens as ranging from 0.004 to 0.013 mm for the five
float glass samples and 0.013 and 0.037 mm for the two nonfloat samples.
This difference between float and nonfloat samples seems to also exist
in flat glass samples; however, the standard deviations for flat glass are lower.
We are, however, unaware of published studies on this.
Survey work18,19 suggests that there is a very high level of discrimination
between glass samples, using thickness notwithstanding, and that most
glasses are sold at nominal thickness such as 3, 3.5, 4 mm, and so on. This
appears to be because there is only a loose agreement between nominal
thickness and actual thickness.
1.9.2 The comparison of color
Large fragments of glass can be compared for color and ultraviolet (UV)
fluorescence relatively straightforwardly by subjective comparison side by
side. For color, samples may be viewed in a variety of lights, and it is often
valuable to use daylight and tungsten light. The background should be
neutral or a complementary color (for instance, the complement of blue is
orange). It is thought that a background of the complementary color
“primes” the eye to observe subtle color differences.
Difficulty is found in expressing the evidential weight of these subjective
comparisons, although they do have value. This is because of the difficulty
in storing subjective color information from samples to produce an estimate
of frequency. Attempts to overcome this difficulty, using visible spectroscopy
and a microspectrophotometer (MSP), suggest limited discrimination (Newton, A.N. unpublished results, 1999).
Small fragments of glass must be treated with great caution with regard
to color and UV fluorescence. It is recommended that all glass examiners
should arrange for themselves to be blind tested. This can be achieved by
producing casework-sized glass fragments from glass of differing colors and
then mixing these fragments into debris. Most examiners will soon discover
that they can barely determine a brown beer bottle fragment from clear glass
or from a strong green wine bottle when the fragments are small, dirty, and
among debris. Differentiating, for example, the pale greens and blues often
used in vehicles is even more difficult.
©2000 CRC Press LLC
1.9.3 Matching edges and matching hackle marks
With large, recovered glass samples, the fragments may first be examined
to determine if coincidental edges exist between the recovered and control
fragments in the form of a physical fit. This is most feasible for large pieces
of glass and is seldom attempted on the typically smaller fragments transferred to clothing, although rare successes are reported. The observation of
these “fracture matches” indicates that both fragments once formed a larger
piece of glass.
Similarly, the edges of a broken fragment will exhibit mechanical markings, known as “hackle” marks (see Figure 1.6), which occur with such
irregularity that a perfect match of an edge from a recovered fragment to an
edge from a source fragment also indicates that both samples have come
from the same source. The successful association by a hackle mark match is
also a rare occurrence in casework and is typically not attempted on fragments recovered from clothing.
1.9.4 Density comparisons
One of the analyses that may be performed for comparison of glass samples
is density. Depending on the information wanted by the forensic scientist,
density can be performed in different ways: the density of the recovered and
control fragment can either be compared directly or both densities can be
determined. Density can be measured at a constant temperature or by varying temperature.20 The gradient of density technique has also been suggested
in literature.21,22 As the latter technique has been shown to be more discriminating, we will present it here. The method most commonly used places a
glass fragment into a liquid of approximately the same density as the glass
(~2.5 g/mL). The density of the liquid is varied by the addition of a miscible
liquid which is more or less dense until the fragment is observed to “float”
in the liquid, not moving up or down in the tube. At that point, the density
of the liquid and the glass are equal, and the density of the liquid can then
be measured. The densitometer can be calibrated with glasses of known
densities. The use of bromoform (CHBr3) as the float liquid in conjunction
with a Mettler densitometer yields a precision of +/–0.0001 g/mL.23 An
alternative to the mix of bromoform and dibromoethane as a float liquid is
an aqueous solution of a nontoxic polytungstate salt. The density of this salt
solution can be easily adjusted by the addition of water, and the water may
be evaporated to concentrate the salt, making it reusable.
Gamble et al.,24 Dabbs and Pearson,25 and Miller16 reported results for
refractive index measurements and density on a set of glass fragments and
noted the strong correlation.
Typical refractive index and density results for casework glass samples
(see Figure 1.8) support these results. Gamble et al. note that “specific gravity
was not as valuable in distinguishing glass samples as was the refractive
index.” However, they do go on to recommend that “it is desirable to deter-
©2000 CRC Press LLC
2.65
Density
2.6
2.55
2.5
2.45
2.4
1.505
1.51
1.515
1.52
1.525
1.53
1.535
RI (Nd)
Figure 1.8 Density vs. refractive index for over 2000 case examples in the U.S.
collected between 1977 and 1990. (Data from FBI laboratory, FBI Academy, Quantico,
VA, Robert Koons, personal communication, 1995, by a densitometer.)
mine as many physical properties as possible.” This suggests that, despite
the obvious correlation, these authors felt that there was still value in determining density if refractive index determination was also performed. Stoney
and Thornton26 have shown that measuring both physical properties may
or may not be useful depending on the error of the techniques.
Improvements in equipment and method, however, have increased the
advantage that refractive index determination has over density, and now
very few laboratories perform both. The small added discrimination is
adjudged not worth the extra cost of analysis.
Interestingly, the correlation between refractive index and density can
be used to refine the density method,27 since the approximate density implied
by the refractive index can be used to prepare a particularly discriminating
density gradient system.
Notwithstanding these advances, the refractive index has largely displaced density as the physical property of choice to be determined. For
example, a recent survey conducted by the SWGMAT* group addressed the
question of added discrimination by density analysis when the refractive
index determination by the Glass Refractive Index Measurement (GRIM)
was performed. One laboratory that had been collecting both refractive index
and density data over many years could show only one example of further
discrimination by density when refractive index was also measured. This
correlation also holds true for refractive index and density ranges above the
usual values for casework.
* Scientific Working Group on Materials (SWGMAT) sponsored by the FBI laboratory, FBI
Academy, Quantico, VA.
©2000 CRC Press LLC
This finding supports the conclusion that the measurement of both
refractive index and density may produce some additional discrimination,
but this is limited by the observed correlation between these two properties.
1.10 Examinations performed on small and large fragments
1.10.1 Recovering glass
A number of methods for detection and collection of glass from clothing
have been reported. These include hanging the garment over a large piece
of paper and scraping the debris with a clean metal spatula onto the paper
for collection; picking with tweezers; using adhesive tape on large surfaces
(or on a surface where fibers are also gathered); in the case of the inside of
a vehicle or a large room, vacuuming the area in question and examining
the filter; and shaking the garment over paper or over a large (previously
cleaned and blanked) metal cone. For garment examinations, each garment
should preferably be packaged and inspected separately. The relative advantages of recovery methods can usefully be debated.
Pounds and Smalldon28 give a comparison of visual searching vs. recovery by shaking. Table 1.1 gives a quick summary of these findings.
Table 1.1
A Summary of the Findings of Pounds and Smalldon28
Size range
0.1–0.5 mm
0.5–1.0 mm
1.0 mm or greater
Recovery by visual searching
0–12%
13–67%
44–100%
Recovery by shaking
72%
85%
93%
Interestingly, when visual searching was performed by two experienced
and two inexperienced examiners, both sets performed similarly. The ranges
in the visual searching column relate to the different performances of the
different examiners and different garment types. These findings suggest that
shaking is an efficient way of recovering glass, especially in the critical size
ranges for typical casework, 0.1 to 0.5 and 0.5 to 1.0 mm. It is noted that the
British view of shaking is a more violent activity than the impression given
by the SWGMAT description of “scrape the debris with a clean metal spatula.” Nonetheless, they both constitute shaking-type methods of collection.
A remaining point of debate is whether to shake over clean paper (and,
if so, how) or a metal cone. The advantages of the cone are seen as a greater
chance of recovery of fragments during violent shaking with the consequent
risk of carry over and contamination. If the cone is to be used, strict anticontamination procedures are typically required, including “blanking”* of
the cone between suspects. The paper method requires that contamination
does not come from the surface under the paper. This is typically achieved
by cleaning this surface and, in some cases, placing a taped down clean sheet
of paper on the work surface before placing the sampling sheet of paper.
* Blanking involves sampling the cone and proving the absence of carried over glass.
©2000 CRC Press LLC
Fuller29 reported recovery rates for various methods of hair combing and
found variable results. Plain combs and combs treated with cotton wool and
Litex were evaluated. No clear trends were discernible regarding the most
efficient type of comb.
1.10.2 Examination of transparent material to determine whether it
is glass
Many examiners begin their career picking up quartz and glass in equal
measure from the debris produced by shaking garments. However, experience tends to allow the examiner to pick up glass exclusively by choosing
those fragments with freshly broken edges and “appearance,” sometimes
referred to as the conchoidal fracturing.
If small particles of clear material are encountered, these can be examined under crossed polarized light conditions to determine if they are isotropic. Anisotropic materials such as quartz or other mineral crystals are
birefringent and rotate plane polarized light. Therefore, they appear bright
under crossed polarizers in at least some orientations. Isotropic materials
such as glass would remain dark in all orientations. Plastic particles can be
distinguished from glass particles by applying pressure on the particle with
a hard point and determining if the material compresses.
The characteristic appearance of quartz in oil can often be recognized
when the refractive index measurement is attempted, and many examiners
do not take the step of determining whether or not the material is glass
before proceeding to refractive index determination. It is hoped that with
experience examiners do not pick up much quartz and if they do the fragments can be detected at the refractive index measurement stage.
1.10.3 The examination of surface fragments
A surprisingly high fraction of casework glass samples recovered from clothing exhibit an original surface, that is a surface that is part of the surface of
the original object.30 This suggests that examination of these surfaces might
be a quick and effective way to examine glass fragments.
Elliot et al.31 point out that examination with a standard stereomicroscope with coaxial illumination allows the examination of specular reflection,
reflections from the surface of the glass. “Examination of the specular reflection image reveals surface details such as pits and scratches…. Curvature
(or lack of it) is revealed by observing the reflected image of, for example,
a probe tip as the probe is moved above the surface. For a flat surface, a
sharp image can be obtained at every point; for a curved or irregular surface
the image is distorted.”31
This method is simple and uses available equipment; however, interference microscopy appears to offer advantages. Elliot et al.31 describe the use
of interference microscopy and their application to surface examination of
glass fragments. Locke and Zoro32 describe a method for performing this
©2000 CRC Press LLC
surface examination using a commercially produced two-microscope system. This methodology has not achieved widespread use despite its utility.
The phenomenon of interference involves allowing monochromatic
light, which has traversed a constant distance (the reference beam), to interact with light that has reflected from the surface of a glass fragment. If we
imagine the fragment to be flat and very slightly angled, then there is some
line on this fragment where the light is reinforcing with the reference beam.
At some distance from this line the glass is either further away or closer to
the source and does not reinforce. Further away still, the distance will again
become an integral number of wavelengths and will again reinforce. The
effect is a series of lines of dark and light. For flat objects these lines are
straight whichever way the glass fragment is oriented. For curved objects
the lines are curved in most orientations.
Locke and Elliot33 pursue a classification approach with much success.
Using their recommendations, glass is classified as flat, curved, or slightly
curved or by suggesting sources such as inner surface of container or tableware,a outer surface of container or tableware,b spectacle lens,c nonpatterned
surface of patterned glass,d patterned surface of patterned glass,e flat glass,f
or windscreen glass.g It can be seen that some of the characteristics described
would apply to two categories; however, blind trials suggest a very high
success rate once experience has been obtained. Error rates are higher for
small fragments. This suggests that classification should be applied to fragments 0.4 mm2, or to sets of three fragments. If sets of three fragments are
to be used, then they should all be <0.4 mm2 in size and they should be
similar in refractive index and interference pattern.
Experience suggests that the technique has discrimination beyond that
suggested previously. The flat sides of different one-sided patterned windows are easily differentiated, for instance, due presumably to the different
surface defects on the flat roller. Even flat glass shows differing amounts of
surface markings from cleaning and may be discriminated in some cases.
Given the cheapness and effectiveness of this technique, it is surprising
that more use is not made of it. It also seems likely that a high fraction of
surface fragments is an indicator of backscattered glass. There is obvious,
but unexplored, potential in this, using the Bayesian approach, to help distinguish between some prosecution and defense hypotheses.
As previously suggested, float glass fluoresces when excited at 254 nm.31
This fluorescence gives the surface a dusty or milky appearance that may
appear faintly yellow or green. This fluorescence is not necessarily visible
a
Smooth curved lines due to the blown nature of the inner surface.
Wavy curved lines due to the mold contact of the outer surface.
c Curved straight lines.
d Straight, but wavy lines resulting from flat surfaces affected by contact with the roller.
e Complex wavy lines.
f Smooth straight lines in all orientations.
g Smooth, very slightly curved lines.
b
©2000 CRC Press LLC
in casework-sized fragments due to their small size and reflections from
broken surfaces.
Lloyd34 reports on experiments examining the fluorescence spectrometry
of glass surfaces of casework-sized pieces. The main emission from a float
surface is a very broad band with a maximum at about 490 nm when excited
at 280 nm. Another excitation occurs at about 260 nm and produces a second
broad emission largely superimposed on the first. A third excitation occurs
at 340 nm, producing an emission peaking at 375 nm.
Nonfloat surfaces show only one strong fluorescence with an excitation
at about 340 nm and an emission peaking at about 375 nm, similar to the
third excitation of the float surface. The excitation at 280 nm is thought to
result from tin. Considerable variation occurs between the fluorescence
intensities for different samples of float glass. The largest variation lies in
the tin fluorescence.
Locke35 describes equipment for viewing the float glass fluorescence on
casework-sized fragments. Inquiries suggest that this equipment is not commercially available. The float surface may also be detected by the use of Xray analysis using the SEM to detect tin.
1.10.4 Refractive index determinations
The dominant physical property investigated in glass has been refractive
index (RI) measurements. RI can provide a high degree of discrimination
between a known and a questioned glass sample.
RI measurements are widely used for comparing forensic glass samples
in crime laboratories. This optical property is easily measured in transparent
materials such as glass and has been measured for more than 60 years with
various methods. As such, it has largely superseded density as the physical
property of choice to be measured.
Figure 1.9 illustrates the change of direction (refraction) that is observed
when light passes from one medium to another. Refraction occurs because
(as described by Snell’s Law in Equation 1.1; also see Appendix A) the
velocity of light in the transparent medium is slowed.
This interaction can be described as (1) the ratio of the wave’s velocity
in a vacuum to the wave’s velocity in the transparent medium or (2) the
ratio of the sine of the incident angle to the sine of the angle of refraction.
RI =
sinθ I
sinθ R
=
VVacuum
VGlass
(1.1)
Becke* first described a phenomenon that he observed while analyzing
geological samples in 1892. Using a microscope to examine rock sections,
Becke observed a bright line inside the edge of a mineral section that had a
* Friedrich Johann Karl Becke, mineralogist, 1855–1931.
©2000 CRC Press LLC
Angle of Refraction
Light
I
R
Angle of Incidence
Glass
Figure 1.9 Refraction of light through a glass medium.
higher RI than its surroundings. Becke also saw that when the objective on
the microscope was raised (focus up), the bright line moved in the direction
of the mineral of higher RI. This Becke line, as it is known today, is a function
of how light behaves at the boundary between two components with different RIs. To measure the RI of glass, one could immerse the glass in oil and
adjust the focus on the edge of the glass fragment. If the objective was raised
and the bright line moved into the oil (and away from the edge of the glass),
then this would indicate that the oil had a higher RI than the glass. One
could then take a miscible liquid such as an organic solvent and add it to
the oil to lower the RI line until it does not move further or disappears. At
that point, the RI of the oil is equal to the RI of the glass. One could then
measure the RI of the oil with a refractometer. This method is very time
consuming and not very accurate.
Around 1930, Emmons described the temperature variation technique,
observing that RI varied with temperature. As the temperature of an oil is
raised, the RI of the oil decreases while not affecting the RI of the immersed
glass to the same extent. Emmons used a hot circulating water bath to heat
the microscope slide and its contents and then waited for the Becke line to
disappear. The water bath was also used to heat a sample of the same oil on
a refractometer so that the RI could be measured once the line disappeared.
This method was an improvement over the previous Becke method, but it
was hard to control the temperature accurately. Therefore, the error was
relatively high.
The double variation method16 involved the variation of both the temperature and wavelength of the light coming through the sample to determine the RIs for three wavelengths. This method has been published as an
accepted method by the Association of Official Analytical Chemists.
Mettler developed and sold a commercial Hot-Stage™ in the early 1970s
that enabled very good control of the temperature of the locally heated
microscope slide and was typically used in conjunction with a phase contract
objective. This method was more accurate than previous methods, but still
very time consuming. The phase contrast objective36 enhances any phase
©2000 CRC Press LLC
contrast caused by retardation or acceleration of the light as it passes through
the glass relative to light that has passed only through oil. This is achieved
by adding or subtracting a large phase difference (typically λ/4) to the small
phase difference existing between those rays which have interacted with
the sample and those which have traversed the surrounding border. This
enhanced phase difference is seen as an optical effect similar to the Becke
line, that is light and dark lines, but is strictly not the Becke method since
that method uses a defocusing which is not required in phase contract
microscopy. The act of defocusing and refocusing becomes progressively
harder as the match temperature is approached and the fragment becomes
less visible.
As noted by Ojena and De Forest,3 “it becomes increasingly difficult to
see the Becke line as one gets closer to the match point of the immersion
liquid and the glass fragment. The contrast of the diffraction pattern (Becke
line) is influenced by the shape and dimensions of the specimen and by the
degree of defocusing of the microscope. These factors, in turn, affect the
accuracy of the match.”
In the mid 1980s, Foster and Freeman developed an instrument which
they called “Glass Refractive Index Measurement” (GRIM) based on the
observation of the glass immersed in oil (Figure 1.10).
Figure 1.10
Glass fragments as viewed from the GRIM II video camera while being
immersed in the Locke B Oil. The fragments on the right are invisible due to the fact
that they are at the “match temperature.”
A phase contrast microscope is used at the fixed wavelength of 589 nm,
and the image is adjusted for maximum contrast. GRIM I and II both use
dark contract phase microscopy. This is where the diffracted rays, those that
have interacted with the sample, are deliberately retarded with respect to
the 0-order (undiffracted) rays.
©2000 CRC Press LLC
Three studies have been conducted to evaluate the accuracy, precision,
and long-term stability of the GRIM method and conclude that the instrument provides for very satisfactory results and should be the method of
choice for the measurement of RI in forensic laboratories.
The GRIM manual suggests that fragments producing edge counts below
ten should not be used or should be repeated manually. Tests by Zoro et al.37
and Coulson, Curran, and Gummer (personal communication) suggest that
there is an increase in the variation of measurement on a single glass pane
with a lower edge count, but the mean remains approximately the same.
This suggests that low edge count fragments may still be used if the added
variance is considered in the comparison and assessment procedure.
1.10.5 Dispersion
RI varies with wavelength, being greater for shorter wavelengths.16 The
standard wavelength for RI determination is the sodium D line. If no superscripts or subscripts are used, then the RI referred to is the Sodium D line
at 25°C. Therefore, N is shorthand for ND25. Other standard wavelengths used
are near the ends of the visible spectrum (typically the hydrogen C [656.3
nm] and F [486.1 nm] lines). The RI at these wavelengths are NC and NF ,
respectively.
There are two ways of expressing dispersion. First, it may be shown as
a plot of RI vs. wavelength or, second, it can be defined as
V=
ND − 1
N F − NC
(1.2)
Locke et al.38 suggest that “for glasses of the type commonly encountered
in casework in the United Kingdom (of Great Britain and Northern Ireland),
dispersion measurements are unlikely to enhance the evidential value” due
mostly to the large observed correlation between dispersion and ND. Locke
et al.38 and Cassista et al.39 dispute the findings of Miller40 and Koons et al.41
who suggested the opposite conclusion. However, in Miller data have not
been given on the discrimination added, and in Koons et al. no phase contrast
has been used, suggesting the possibility of low precision.
1.10.6 Refractive index anomalies
It has been known for some time that the float surface of float glass has a
different RI to the bulk of the glass. Underhill42 describes the origin of the
multiple RI phenomena. The float surface is enriched in SnO2, and the opposite surface appears to be depleted in various other constituents and, consequently, enriched in SiO2. The float surface typically has a lower RI to the
bulk. Underhill42 and Davies et al.43 raised the exciting prospect that a separate float RI might be measurable and, hence, an additional aspect of dis-
©2000 CRC Press LLC
crimination might be available. Later research37 confirms the presence of RI
anomalies at the float surface, but suggests that this RI is approached
smoothly over the last few microns of depth and, therefore, a discrete surface
RI does not exist. They also show that the RI of the antifloat surface (the
surface opposite the float surface) has an anomaly in the other direction (that
is toward higher RI). This antifloat anomaly is smaller than the float anomaly
in most, but not all, of Zoro et al.’s samples.37
Nonfloat flat glass may also on occasion exhibit surface anomalies. Two
of Zoro et al.’s nonfloat samples showed anomalies, in both cases toward
higher RI. One sample showed this at one surface and another at both
surfaces.
Container glass and tableware may also show surface anomalies (in this
case in either direction).
These findings do not absolutely exclude the use of “surface RI” as a
discrimination tool, but do suggest that this use will be very difficult. It is,
for instance, theoretically possible to consider “maximum” deviation as a
discrimination tool. In order to do this it will be necessary to identify fragments expected to show maximal deviations, possibly by confirming that
the measured edge is truly surface (and reasonably thin but measurable).
The authors are aware of no laboratories applying this technique.
1.10.7 The examination of tempered (toughened) glass by annealing
Locke et al.44 describe an annealing process where a specifically designed
oven is used to anneal the small fragments typically recovered from clothing.
Locke and Hayes45 report the results of this technique when applied to a
number of different glass sources.
Table 1.2 is a summary of Locke and Hayes findings; positive ∆RI values
mean that the RI was higher after annealing.
Table 1.2
A Summary of Locke and Hayes’45 Findings
Glass type
Tempered specimens
Nontempered float, patterned, or plate glass
Container glass
∆RI
0.00173–0.00206
0.00086–0.00144
0.00073
Table 1.2 suggests that tempered specimens can be differentiated from
nontempered specimens. Later work46 on 200 random survey items suggested that an ∆RI value above 0.00120 identifies the glass as toughened,
while values less than 0.00060 identify the glass as not toughened with an
inconclusive region in between (please note that these values do not follow
from Locke and Hayes’ work which would suggest a value of ∆RI = 0.00150).
The studies performed by Cassista and Sandercock,39 Locke et al.,44,45,47,48
Winstanley and Rydeard,49 and Marcouiller50 confirm that toughened glass
can successfully be classified using annealing.
©2000 CRC Press LLC
This approach is the traditional classification approach. A different
approach is possible. The ∆RI value per se is an element of discrimination
and can be used as that rather than as an element of classification. This
point is a theme in glass where much effort has been expended in classification type investigations where the discrimination type approach is as
valid if not more so. In the case of annealing, two types of discrimination
information are available. The first described earlier is the ∆RI value, and
the second arises from the fact that the variance of RI in annealed glass is
less than in unannealed glass. Hence, the comparison of two sets of
appealed measurements is expected to give more discrimination (or a
more peaked density, see later).
1.11 Elemental composition
Elemental analysis can either be used for classification or to discriminate
glasses of the same type (windows, container, etc.) having similar physical
characteristics. Indeed, RI differences correspond to small compositional
differences in the major elements silicon (Si, ~30%), sodium (Na, ~8%),
calcium (Ca, ~8%), magnesium (Mg, ~2%), and potassium (K, ~1.5%). It is
possible, however, to have two different glass samples and be able to observe
the same density and RI, and measure a difference in the minor element
composition of aluminum (Al, ~1%) and iron (Fe, <0.3%), or by trace element
content of barium (Ba), manganese (Mn), titanium (Ti), strontium (Sr), and
zirconium (Zr). Ba, Mn, Ti, Sr, and Zr are all usually present in concentrations
of less than 0.1%.
The techniques used for classification and discrimination may be different, as the determination of glass type depends on the major elements,
whereas discrimination is mostly based on minor or trace elements. Numerous methods have been proposed for the elemental analysis of glass, such
as SEM-EDX,51-53 X-ray fluorescence (XRF),54-57 neutron activation,58 or inductively coupled plasma (ICP) techniques.41,59,60 Many different techniques for
the analysis and comparison of glass samples have been used because in
many cases forensic laboratories must often use the available instrumentation and adapt its use to as many types of evidence analyses as possible. A
review of some of the methods used for the elemental analysis of forensic
glass samples was reported by Buscaglia61 who discussed the advantages
and disadvantages of the individual analytical techniques.
Although each of these techniques has its strengths, each is limited in
practical usage by one or more of the following: sample size requirements,
whether or not the technique is destructive, sensitivity, precision, multielement capability, or analysis time.
It has been argued that the last four decades have seen a change in the
glass manufacturing industry as it greatly improved the quality control
within glass plants due to the introduction of the float process for glass
manufacturing. This improvement in the quality of the glass product has
©2000 CRC Press LLC
been further enhanced by computer-managed processes that also allow manufacturers to control the physical and optical properties of the glass such as
thickness and RI to a great degree. In addition, the methods and formulation
among manufacturers and plants around the world has become more uniform. This has caused a noticeable decrease in the variation of RI for the
population of flat glass produced by any one plant. While this trend appears
to have been at least partially offset by the growing internationalization of
the market, leading to more plants selling into any one market, there have
been persistent concerns about a general loss of overall discrimination if the
sole method used is RI. This has led to a critical review of other methods of
analysis that could add discrimination.
A second school of thought considers elemental analysis as a tool to be
used when required. Therefore, depending on the circumstances, there may
be no need for elemental analysis or there may be a need for a technique
that adds a little more discrimination or a very powerful technique. The
scientist will not only consider the discriminating power of the technique
when choosing if and how elemental analysis should be performed, but
she/he will consider the circumstances of the case, the cost, the destructive
nature of some techniques, and the availability of equipment. It is in this
context that we would like to present the techniques used nowadays in
forensic laboratories or in research: X-ray and ICP techniques.
1.11.1 X-ray methods
Two techniques have been proposed for the analysis of glass: SEM-EDX and
XRF. Both are considered to be rapid, relatively sensitive, nondestructive,
and complementary, as SEM is more sensitive to small atomic number elements (major and minor elements in glass). These techniques have both been
used for classification and discrimination.
1.11.1.1 Classification of glass using X-ray methods
Keeley and Christofides56 showed that magnesium and iron allowed the
differentiation of windows from other glass types. These results have been
confirmed by other researches using SEM-EDX, such as Ryland.53 Ryland
used both SEM and XRF, as the latter is more sensitive to iron. In this
research, it was found that old windows contained low levels of magnesium,
but that the high iron content still allowed differentiation from container
glass. Terry et al.52 used sodium, aluminum, silicon, calcium, and potassium
in addition to magnesium. The content of potash and lime allowed the
differentiation of borosilicate labware from borosilicate headlamps; the content of potash (in excess of 5%) differentiated spectacles and lead glasses.
Flat glass was identified by its low aluminum content. Howden et al.54
successfully used XRF to classify glass; the elements used were sodium,
magnesium, potassium, calcium, iron, arsenic, barium-titanium, strontium,
and zirconium.
©2000 CRC Press LLC
For additional information on classification schemes showing differences
between container and float glass, the reader is referred to studies conducted
by Hickman.59,62,63
1.11.1.2 Discrimination of glass using X-ray methods
XRF techniques have also been reported to discriminate glass of the same
type, having indistinguishable RI and density. In the first study published
on the subject, Reeve et al.51 reported that out of 81 glass samples, 2 were
indistinguishable. In a similar study, Andrasko and Maehly64 were able to
distinguish all but 2 of 40. With emission spectroscopy a further pair was
discriminated. Studies on the discriminating power of the X-ray methods
show that XRF may be more discriminating than SEM, but less discriminating than ICP techniques.
Rindby and Nilsson57 have reported the use of micro-XRF. This technique
as well as total reflection XRF (TXRF) seems promising.
1.11.2 ICP techniques
Inductively coupled plasma-atomic emission spectrometry (ICP-AES) has
been applied extensively to trace element analysis of glass.41,59,62,65-69 ICP-AES
combines multiple element detection capability with good sensitivity, small
sample size requirements, and a large linear dynamic range, making it well
suited for the trace element analysis of glass.
1.11.2.1 Classification of glass using ICP techniques
ICP-AES has been proposed as a classification and discrimination method.
Hickman59 reported a scheme for classifying glass samples as sheet, container, tableware, or headlamp. RI, as well as elemental concentration (manganese, iron, magnesium, aluminum, and barium), allowed the classification
of glass samples 91% of the time. In 1988, Koons et al.66 were able to classify
182 sheet and container glasses correctly, with the exception of 2 windows
classified as container and 2 containers classified as sheets. In most instances
it was even possible to correctly associate each sample with a manufacturing
plant.
Almirall et al.60 have also shown the use of classification methods for
glass sources with discriminant analysis routines using elemental data by
ICP-AES.
1.11.2.2 Discrimination of glass using ICP techniques
Early work by Hickman et al. in the U.K. provided a basic set of elements
which allowed a high degree of discrimination between glasses.59,62,70 These
elements included lithium, aluminum, magnesium, manganese, iron, cobalt,
arsenic, strontium, and barium. Investigations by Koons and co-workers in
the U.S. have compared ICP-AES to RI and energy dispersive XRF and
demonstrated on a sampling of 81 tempered sheet glasses a much higher
©2000 CRC Press LLC
degree of differentiation for ICP-AES.41 The elements measured were aluminum, manganese, iron, magnesium, aluminum, barium, calcium, strontium,
and titanium. Both ICP-AES and RI were also studied extensively by Almirall
in his Ph.D. thesis. Wolnik et al.67 reported that the use of ICP-AES for the
discrimination of container glass using barium and magnesium concentration proved to be useful to discriminate manufacturers.
A more sensitive and potentially more informative method of glass
analysis is inductively coupled plasma-mass spectrometry (ICP-MS).71,72
Analytical applications of ICP-AES and ICP-MS are similar, while ICP-MS
usually affords 10 to 100 times the sensitivity of ICP-AES, allowing either
ultratrace element detection or the analysis of smaller samples. ICP-MS also
provides isotopic information that is useful in isotope dilution analyses,
which improves both accuracy and precision.73
The first reported application of ICP-MS to forensic glass samples was
made by Zurhaar and Mullings73 who analyzed seven glasses with identical
RIs. Due to the sensitivity in ICP-MS, samples as small as 500 µg were
analyzed with detection limits below 0.1 ng/mL. While 48 elements could
be accurately determined with relative standard deviations of less than
4%, it was found that a suite of 25 elements produced successful discrimination levels in 85 to 90% of the glasses tested.
A more extensive investigation of ICP-MS analysis of glass has recently
been published by Parouchais et al.74 Sample digestion methods were compared, and up to 62 elements were determined in a range of glass samples.
Successful differentiation of glasses of similar RI was accomplished by comparing element concentrations and element ratios (e.g., Sr/Ba). The latter
technique is useful in that samples need not be weighed, thereby simplifying
the analysis procedure. The use of elemental ratios is appealing because they
are not dependent on mass, not subject to instrument drift, and are less prone
to signal suppression effects. However, they can be subject to temporal
variations due to changes in mass bias and plasma temperature, making
database development difficult.
An Australian group (personal communication, the National Institute
for Forensic Science, NIFS) and the BKA (BundesKriminalAmt) have conducted research on the application of laser ablation (LA)-ICP-MS. This
method has the great advantage of being fast and nondestructive.
These studies have demonstrated that trace element analysis by ICPAES and ICP-MS is a powerful method for discrimination of forensic glasses
when more common techniques, such as RI measurement, fail to discriminate.
Interpretation of elemental concentrations, whether determined by XRF,
ICP-MS, or LA-ICP-MS, requires a fundamental knowledge of the variation
within a population. The fundamental knowledge of these distributions is
not presently available for ICP-MS. It is expected that current research in the
U.S. and Germany will take the elemental analysis of glass by ICP methods
from the research laboratory to the operational crime laboratory and hence
into the courtroom.
©2000 CRC Press LLC
1.12 Summary
We have presented a sampling of techniques available to characterize glass.
The main thesis of this book is how to interpret the results produced. From
here onward, we will be presenting both the “classical approach” to interpreting this data and the “Bayesian approach,” which we advocate. It was
thought necessary here to introduce the type of data that may be developed
on glass so that some, at least, of the subtleties might be understood.
1.13 Appendix A — Snell’s law
The relationship described by Snell’s law is easily derived. Construct the
triangles shown by dropping perpendiculars. Note that the angles marked
are θi and θr by using the concept that internal angles of the triangles are θi,
90, and 90 – θi and θr, 90, and 90 – θr, respectively. Then,
sinθ i =
x
h
(1.3)
sinθ r =
y
h
(1.4)
sinθ i x
=
sinθ r y
(1.5)
and, therefore,
and x/y are in the ratio velocity in a vacuum/velocity in the medium.
Vacuum
x
θi
90-θi
h
y
θr
Medium
©2000 CRC Press LLC
chapter two
The conventional approach
to evidence interpretation
Let us assume that the police have taken a sample of size n control fragments
of glass from the crime scene and that the laboratory analyst has recovered
m fragments from the suspect’s clothing (and hair). A single RI measurement
has been made for each fragment using the standard immersion test.3 Therefore, we have n measurements of RI on the control glass x1, x2,…, xn and
measurements y1, y2,…, ym on m fragments recovered from the clothing (and
hair). Because of internal variation and the error of the technique, even if
the control and recovered glass have the same origin, there will be differences
between the two sets of measurements. Therefore, it is necessary to determine if these differences are due to internal variation and the error of the
measurement technique or to the fact that the fragments come from different
sources.
How should the problem of comparing these two sets of measurements
be approached? At one extreme there is the view that an experienced forensic
scientist can effectively do this purely by personal assessment. The other
extreme calls for an objective statistical test giving a match/no match result
quite independent of human judgment. Inevitably, neither extreme is sustainable in practice. The “personal judgment” school is open to the criticism
of being unscientific unless the persons who make the judgments can show
a convincing level of performance in tests designed specifically for the purpose of establishing competence in this activity. The “objective test” extreme
follows a common illusion, regrettably fostered by many statisticians, which
ignores the fact that no test can be formulated without an underlying theoretical model. The validity of such a model in any given case must be
determined by personal assessment, taking account of information other
than that contained in the measurements alone.
The optimal path must fall somewhere in between these two extremes.
The central role of expert judgment is not denied, but it is recognized that
it cannot be unfettered. It should be augmented and circumscribed by systems for ensuring competence. Neither can the enormous advances that have
©2000 CRC Press LLC
been made by statisticians in developing methods for use in circumstances
such as these be ignored.
The next section follows the development of statistical tools in glass
evidence interpretation from a historical perspective. This chapter will start
from the objective test extreme, but will later develop to show how a more
optimal approach may be achieved.
2.1 Data comparison
The comparison of RI/elemental measurements has been done in various
ways. There are very simple tests such as the “range test” or the “±3SE rule”
and some more complex tests such as the Student’s t-test, the Welch test, or
Hotelling’s T2. These methods all test the hypothesis H0 that the recovered
and control fragments have the “same” physical or chemical characteristics.
However, we know that there will often be cases in which the recovered
fragments have come from two or more sources. Therefore, before testing
H0, we could adopt one of two policies. The first, and oldest, approach is to
test each recovered fragment against the control mean. This approach has
its merits, but these are outweighed by two problems: (1) each single test is
comparatively weak, and (2) the potential for drawing an incorrect conclusion increases with every test. A more powerful approach is to adopt some
kind of preliminary procedure for identifying groups among the recovered
fragments. The conventional methods — illustrated for RI measurements —
can be schematized as shown in Figure 2.1.
It is perhaps surprising that there are so many different methods for the
interpretation of glass evidence. Therefore, it is logical for one to wonder
which test should be chosen and whether or not the recovered fragments
should be grouped. To aid this process we will present the advantages and
disadvantages of the methods.
2.1.1
Range tests and use of confidence intervals
The range test is a nonparametric test (the expression “nonparametric”
means that the test design makes no assumptions about the underlying
distribution of the data). It has the great advantage of being a computationally simple test. One may accept (or fail to accept) the hypothesis that the
recovered and control fragments have the same physical or chemical characteristics if the measurement made on each recovered fragment lies within
the range defined by the maximum and minimum values observed in the
control sample. The mathematical relationship can be formalized as follows:
Reject H 0 if min xi > min y j or max xi < max y j
i =1…n
j =1…n
Otherwise fail to reject H 0
©2000 CRC Press LLC
i =1…n
j =1…m
Control fragments X
Recovered fragments Y
RI
RI
Measurements on n fragments y1, y2, .., ym
Comparison with X one
fragment at the time
Measurements on m fragments x1, x2, .., xn
Grouping
Determination of the
mean and SD of X
Group 1, 2, ..., k
yi is compared to
the range of X
yi is compared
to X + 3SE
Student's t-test or Welch test
M
e
a
s
u
r
e
m
e
n
t
s
C
o
m
p
a
r
i
s
o
n
o
f
M
e
a
s
u
r
e
m
e
n
t
s
Match/nonmatch
Match
Interpretation
Coincidence probabilities
Figure 2.1
I
n
t
e
r
p
r
e
t
a
t
i
o
n
Discriminating power
Conventional approach to glass evidence.
The real disadvantage of this method is that it has an unacceptably high
false acceptance rate, i.e., it will imply two samples are the same when they
are often not.
2.1.2
Confidence interval
The use of the confidence interval (also called of the ±2 or ±3SE rule) presupposes that the measurements are normally distributed and that the variance of the recovered glass is the same as the control glass. The test is as
follows: if the measurement of the recovered fragment falls within the con-
©2000 CRC Press LLC
fidence limits determined by the control sample, then the hypothesis H0 that
the recovered and control fragments have the same physical or chemical
characteristics is accepted. Otherwise, the fragment is said to be different.
The mathematical relationship can be formalized as follows:
[
]
H 0 accepted if : y j " x # tn#1 (! 2)se( x )$j = 1, … m
Otherwise reject H 0
where tn–1(!/2) is the 100(1–!/2)% critical value for the t-distribution on n–1
—
degrees of freedom, and se(x ) is the standard error of the mean.
The significance level* (!) can be set at any level, but is usually one of
5, 1, or 0.1%.** There is a considerable amount of arbitrariness about the
choice of significance level. Why should we choose any particular significance level — 5, 1, or 0.1%, or any other? Let us leave this issue to one side
for the time being and proceed with a 1% significance level. By setting 1%
as the desired significance level we accept that, in theory, on average 1% of
cases in which the control and recovered fragments truly came from the
same source we would incorrectly reject the null hypothesis. That is we
would mistakenly conclude that the recovered fragments had not come from
the window at the scene. The significance level is also known as the Type I
error rate, the theoretical probability of a false negative. If we were dissatisfied with this false negative error rate, then we could change to a 0.1%
significance level. However, the consequence would be that we would be
less likely to discriminate in those cases in which the control and recovered
fragments had, in fact, come from different sources. That is we would
increase the Type II error rate or the false positive rate. Thus, the significance
level is inevitably a compromise between Type I and Type II errors.75
Two rules that seem to avoid setting the significance level are the ±3SE
rule (see, for example, Slater and Fong20) and the ±2SE rule.62 However, these
rules still use a significance level. The problem is that it changes with respect
to sample size. For example, with five control fragments the Type I error rate
would be about 4% for the ±3SE rule and 11.6% for the ±2SE rule. With 20
control fragments these figures would be 0.7 and 6%, respectively. Compared
to the range test, using the rule of ±3SE presents about two times less false
negatives, but three times more false positives.
* Most statistics tests such as Student’s t-test or the Welch test presented later presuppose that
the significance level is fixed.
** The significance levels referred to here are “theoretical.” They will be correct if a number of
assumptions are satisfied. In practical glass work there may be a considerable difference between the theoretical significance level and that achieved in practice. The consequence of this
is that the Type I error rate (false negatives) is typically larger than the theoretical Type I error.
Some effort has been made to identify which of the typically made assumptions are being
violated in practical glass work, and, while there are some candidates, there is not yet international agreement on this matter.
©2000 CRC Press LLC
It is very important to note that these error rates are per test. Each time
a fragment is compared to the confidence limits of the control sample there
is a probability ! of drawing an incorrect conclusion. Therefore, the probability of making at least one incorrect conclusion if we are comparing m
fragments is approximately m!. For example, with a significance level of 1%
and ten recovered fragments the overall Type I error rate is approximately
10%, i.e., ten times higher than desired.
2.2 Statistical tests and grouping
As the tests presented in this section assume that groups and not single
measurements are compared, the different articles on grouping will be presented before we proceed with the statistical tests.
2.2.1 Grouping
There are various ways in which grouping can be carried out.76 These can
be divided into the agglomerative method of Evett and Lambert75 and the
divisive method of Triggs et al.77,78
Any thorough statistical analysis must allow for the possibility that
recovered fragments may have come from different sources. Current
treatments7,8 of forensic glass evidence require prior knowledge of the number of groups of glass on the clothing.
Conventional statistical clustering or grouping techniques rely on large
sample sizes to achieve their results. In a forensic examination, although the
number of recovered fragments may be high, it is very uncommon for more
than 20 of these to be measured. Thus, conventional techniques are not
feasible. It is desirable to have the optimal approach for identifying groups
within the recovered sample. This section will explain the divisive method
of Triggs et al.77,78 for detecting groups and demonstrate its efficacy against
the currently used agglomerative methods.
In the following section we assume that a sample of m fragments of glass
has been recovered and their RIs y1, y2,…, ym have been determined. Furthermore, we assume that each fragment is drawn from one or more distinct
sources. However, the true source of each fragment is unknown. The grouping problem is equivalent to assigning each fragment to a group so that those
within a group have similar RIs.
2.2.1.1 Agglomerative methods
The agglomerative solution to this problem is a “bottom-up” approach.
Initially, the fragments are considered to have come from m different sources
or belong to m distinct groups. The agglomerative algorithms attempt to
classify the fragments as having come from fewer than m sources by considering the effect of a fragment on the range of a particular grouping of
fragments. If this range is too large, the fragment is put into a separate
group.75
©2000 CRC Press LLC
Evett and Lambert 1 (EL1)
The m RIs are ordered, and the median RI is identified along with its nearest
neighbor. If these two fragments are close enough together, i.e., if their range
(the largest RI minus the smallest RI) is less than some critical range r(!;2),
then they are assumed to have come from the same source. The next closest
neighbor is selected, and again the question of whether it is close enough,
in terms of range of the three fragments, is asked. If the fragment is close
enough, then it too is classified as having come from the same source. This
process continues until either all the fragments belong to the same group or
a fragment is too far away. If a fragment is too far away, then the process is
repeated on each remaining set of ungrouped fragments. If there is only one
fragment under consideration, i.e., all other fragments have been grouped,
this fragment is said to have come from a different source and the process
stops. EL1 is covered in more detail in Appendix A.
The critical values, r(!, m), are scaled standardized normal ranges. The
scaling factor used by Evett79 is a weighted estimate of the unknown standard
deviation obtained from 230 control samples. The critical values are listed
in Table 2.5 in Appendix B.
In subsequent analysis, Evett and Lambert75 assume that the recovered
samples come from normal distributions. It seems sensible then to use percentage points from the empirical sampling distributions of the range of a
sample of size m from a normal distribution rather than the ranges used by
Evett.79 These values are listed in Table 2.6 in Appendix B. This modified
algorithm is denoted Evett Lambert Modification 1 (ELM1) and has one
additional change in that it checks the range of the whole sample against
the particular critical value before stepping into the regular EL1 scheme.
ELM1 is described in more detail in Appendix A.
A further modification to ELM1 is proposed with an alternative starting
procedure to avoid the possibility that the algorithm may start on the edge
of a group. This modified algorithm is denoted ELM2 and is detailed in
Appendix A.
2.2.1.2 Divisive methods
The divisive solution to the grouping problem is a “top-down” approach.
Initially, the fragments are considered to have come from one source or all
belong to one group. The divisive algorithms consider the differences
between the fragments’ RI and whether partitioning the fragments into two
separate groups would increase the homogeneity within groups.
The modifications of Triggs et al.76-78 to the divisive method of Scott and
Knott80 gave rise to the following algorithm.
Scott Knott Modification 2 (SKM2)
The m RIs are ordered, and m–1 different groupings are considered. The
distinctiveness of a particular grouping is assessed by comparing the
summed squared distances of each fragment from the mean of the group to
©2000 CRC Press LLC
which it is assigned to the squared distance between the group means.
Initially, we assume that the RIs of the recovered fragments are normally
distributed with common variance s2 and that we have an independent
estimate, s02, of s2. The maximum summed squared distance is scaled by s02
to give %* and compared to a critical value, %*(!;m) If %* > %*(!;m), then the
fragments are divided into two separate groups, and the group membership
is provided by the particular configuration that gave %*. If a division was
made, then the process is repeated for each of the two subgroups until no
further divisions can be made. The algorithm SKM2 is covered in more detail
in Appendix A.
The critical values %*(!;m) are found by simulation. Assuming that all m
fragments come from one source, 10,000 random samples of size m are drawn
from a normal distribution with mean 0 and s2. For each sample, the value
of %* is calculated. The 10,000 %* values were sorted and %*(!;m) is the 10,000
× ! largest value, e.g., for m = 10 and ! = 0.05 the 10,000 × 0.05 = 500th
largest value of %*, which is 17.54. This is taken as the critical value. We
expect that when the ten fragments all come from a single source, the resulting %* will exceed %*(0.05;10) = 17.54 in only 5% of cases.
The critical values %*(!;m) for a given m and ! are listed in Table 2.4 in
Appendix B. These methods can be best demonstrated by the following
example.
Example
Assume that a sample of 11 fragments is recovered from the suspect’s clothing and the RIs have been determined. The fragments are sorted into ascending order and have been labeled 1 to 11. It is assumed that the recovered
fragments come from three different sources.
The divisive algorithm proposed by Triggs et al.76-78 consists of a
sequence of partitions of the data into two groups based on a maximal
between group sum of squares (BGSS). In Figure 2.2 the divisive procedure
considers the m–1=10 possible partitions of the ordered data into two groups.
The BGSS is calculated for each possible partition, and the maximum BGSS
is found. This maximum occurs when the data is partitioned into a group
containing fragments 1 to 8 and a second group containing fragments 9 to
11. This procedure is repeated for each subgroup until no further partitions
can be made. The groups are taken to be the terminal nodes or leaves of the
tree in Figure 2.2. In some sense the divisive algorithm can be considered a
top-down approach.
The agglomerative algorithm on the other hand works from the bottom
up. The median of the fragments is determined, and the fragment with the
closest RI to the median is selected as the first element of the “group.”
Neighboring fragments are then added into the group until their range
exceeds tabulated critical values for the range of a normal sample. In Figure
2.3, the median is fragment 6. The next fragment closest to 6 is 7. This is
added to the group as the range of this group of size 2 is less than the scaled
0.95 quantile of the distribution of the range of a sample of size 2 from normal
©2000 CRC Press LLC
1,2,3,4,5
1,2,3,4,5,6,7,8
1,2,3,4,5,6,7,8,9,10,11
6,7,8
9,10,11
Figure 2.2
Divisive grouping.
1 2 3 4 5 6 7 8 9 10 11
Figure 2.3
Agglomerative grouping.
distribution. Fragment 3 is added next as the range of the group is less than
the 0.95 quantile of the distribution of the range of a sample of size 3 from
normal distribution, and so on until the procedure attempts to add fragment
6 to the group. The range from fragments 1 to 6 is too big for a sample of
size 6, so the procedure is started again on the subgroup formed by fragments
6 to 11. This procedure is repeated until all the fragments have been grouped.
Triggs et al.77,78 show that the divisive approach gives better detection probability on the basis that the divisive approach makes more efficient use of
the evidence in the small sample size situation.
In some further research, Curran et al.76 have shown the impact of using
automatic grouping algorithms in a Bayesian interpretation of glass evidence.
2.2.1.3 Performance
From the work of Triggs et al.77,78 there seems little question that the performance of the divisive method (SKM2) exceeds the alternative agglomerative
©2000 CRC Press LLC
method (EL1). SKM2 also consistently outperforms the modified agglomerative methods (ELM1 and ELM2). The rates of misdetection of groups by
SKM2 for very small samples are slightly high, but due to the lack of information in these cases this is not unexpected. These extensive computer
simulations, which test the efficacy of the various methods, suggest that
SKM2 must be advocated in place of grouping algorithms based on EL1.
An alternative candidate to these methods is subjective grouping “byeye.” It is difficult to make scientific comments on the effectiveness of this
method as no tests have been reported and it is difficult to see how they
could be performed. They do have the advantage of needing no computing
power. While subjectivity is not, per se, wrong, it should be weighed carefully against objective alternatives.
If subjective methods are employed, there appear to be two associated
issues that may be discussed.
First is the dot diagram method. If the RIs of the recovered fragments
are plotted graphically, this aids “by-eye” grouping. It seems desirable to
always use the same scale so that experience hones the “eye.”
Second, it seems wise, but not always possible, to cover the control group
when grouping. This is easy if there is only one “cluster” of values and has
the obvious advantage of removing the possibility of bias. When there are
multiple or overlapping clusters, the “blind” grouping might be meaningless. A plausible alternative might be to pursue reasonable “prosecution”
groupings and reasonable “defense” groupings as two viable alternatives.
2.2.2
Statistical tests
2.2.2.1 Hypothesis testing
Formal statistical hypothesis testing of data developed at an enormous pace
from the 1920s, mainly through the work of R.A. Fisher who established a
methodology that rests on the concept of a “null hypothesis.” This concept
is used in the glass problem, employing the “grouped” approach, in the
following way. If the two sets of glass fragments, control and recovered, have
come from the same window, then the measurements will all be random
variables with the same statistical distribution — this forms the null hypothesis. If we know what sort of statistical distribution is appropriate, then we
can use relatively straightforward mathematical methods to predict the
behavior of certain functions (called “test statistics”) of the x and y. It is
necessary to make some assumptions in order for the behavior of the tests
to be guaranteed. These assumptions should be kept as explicit as possible.
We will discuss two hypothesis tests that are both very similar to each other.
These have both been offered in the glass context and implemented in casework. They are the Student’s t-test and Welch’s modification to the t-test.
Both are standard statistical tests and appear in standard texts and statistical
packages such as Minitab®, State College, PA; SAS®, Cary, NC; and
Microsoft® Excel, Redmond, WA (for instance, in Excel the two-sample equal
variance option corresponds to what we will describe here as the t-test and
©2000 CRC Press LLC
the two sample unequal variance option corresponds to Welch’s modification
of Student’s [Gosset’s] original work). Here, we discuss the assumptions
made by these tests.
A1(a): The control and recovered measurements x1,
x2,…, xn and y1, y2,…, ym have the same mean; and µx
= µy are representative of the distribution of measurements among the fragments which came from the
scene window (or broken control object).
This assumption is untested. The recovered and control fragments may
be different in terms of their underlying means. In such a case both tests
discussed here will be more likely to result in a rejection rate higher than
the chosen nominal 1%. Factors that have concerned us include an inference
that may also be made from experimental work.47 Locke and Hayes47 show
that the center of one particular windscreen examined in detail had a different sample mean RI to the edges. It is quite plausible to suggest that in a
hit-and-run case the middle of a toughened glass windscreen is left on the
road so that only the fragments from the edges can be taken as the control.
In such a case µx & µy. No viable alternative is known at this time. Such an
effect should elevate false negatives, but not false positives; therefore, this
assumption may, nonetheless, be reliable.
A1(b): sx and sy are estimates of the same parameter ',
the population standard deviation.
This assumption asks whether the two samples both give equally valid
estimates of the population parameter '. It is initially difficult to see why
this should not be true, and most early work made this assumption. However, theoretical and experimental considerations suggest that in some cases
the recovered group has a larger underlying variance than the control. This
may be because they are harder to measure, choice is removed, or because
they are “loaded” with surface or near surface fragments. Locke and Hayes47
observed that most, but not all, of the variance in a window exists across
the depth of a single piece.
These authors studied (inter alia) the variation in RI across a toughened
windscreen. Three areas, A, B, and C, were sampled across the diagonal of
the windscreen. Therefore, there are three possible t-test comparisons that
can be made between these areas. The data from Locke and Hayes’ work
are presented in Table 2.1.
The t-values are 2.98 for A vs. B, 4.04 for B vs. C, and 1.41 for A vs. C.
Therefore, of the three possible comparisons, two would fail a t-test at the
99% significance level (t48(0.005) = 2.68). This suggests that a representative
control is necessary. Often in practice a very small control is submitted and
even a large control sample may be exclusively from just one part of the
window. This problem is likely to be less for untoughened float glass, but
©2000 CRC Press LLC
Table 2.1
Area
A
B
C
The Variation Across a Single Windscreen
Number of
measurements
25
25
25
Mean
1.51674
1.51682
1.51670
SD
9×10-5
10×10-5
11×10-5
the doubt must still remain as to whether the control submitted is a representative sample.
In a case where a small control is received, it is plausible that a slight
underestimate of 'x is obtained.
We offer two approaches: one that makes assumption A1(b) (Student’s
t-test) and one that does not (Welch’s test). It would seem reasonable to make
assumption A1(b) and proceed with the t-test whenever the samples are of
equivalent quality. This would occur, for instance, in a hit and run. Large
fragments of glass have been recovered from a roadway and may be compared to similar quality fragments samples from the vehicle or from a ram
raid where large pieces of glass are recovered from a vehicle (remember we
are discussing the variance effects here not the means). However, in the
majority of cases where a clothing sample is compared to “clean” glass from
a control sample, we doubt that A1(b) is valid and we recommend proceeding with Welch’s test.
A2: Measurements on glass fragments from a broken
window have a normal distribution.
We are not aware of any experimental validation of this assumption and
this is a serious omission in glass examination. There are obvious situations
where this assumption breaks down, particularly because of surface effects.
It is a standard belief that scope for failure of the assumption in a manner
which would have a dramatic effect on the test is limited, and this has been
confirmed by computer simulations.81
A3: The recovered fragments have all come from the
same smashed glass object.
When we make assumption A3, we are assuming that we have first
employed some sensible grouping algorithm and are comparing a group of
recovered fragments with a group of control fragments. This assumption
troubles many experienced examiners with whom it has been discussed.
Evett and Lambert75 argue that the advantages considerably outweigh the
disadvantages. They demonstrate the benefits of a grouping/t-test combination against a range test approach concentrating on the false positive rate
(which, as previously stated, is larger with the range test). A full Bayesian
analysis may not require the grouping assumption; however, we are unable
to do more than imagine how this might proceed at this time. We believe
©2000 CRC Press LLC
that at some future time grouping will be seen as an interlude in glass
examination. However, at this time the evidence strongly favors it.
2.2.2.1.1 Student’s t-test. The Student’s t-test is used to make inferences about a population mean using the information contained in the sample mean. In general, a hypothesis test follows a standard framework.
1.
2.
3.
4.
5.
6.
Ask a question.
Propose a null hypothesis.
Propose a mutually exclusive alternative hypothesis.
Collect data.
Calculate the test statistic.
Compare this test statistic to values that would be expected if the null
was true.
7. If the test statistic is “too large,” reject the null hypothesis.
8. Answer the original question.
In the context of evaluating forensic glass evidence, the null hypothesis
is always the same. Specifically, we hypothesize that the recovered and
control glass are two samples from the same window. Mathematically, we
are testing the hypothesis H0: µx = µy. An equivalent and more prevalent
reformulation of this hypothesis is H0: µx – µy = 0.
If the null hypothesis is true, then it is expected that the difference
between the means of the two samples will be small compared to the differences observed between samples from distinct sources. In this case
“small” is defined with respect to some measure of the variability of RI
within a window (usually the variance or standard deviation).
The comparison of the means of two random samples from distinct
normal distributions is treated in many elementary textbooks on statistics.
The problem can formally be stated as the following:
We have two independent random samples x1, x2,…, xn and y1, y2,…, ym
which are assumed to be independently and identically distributed (iid) as
normal random variables. That is xi ~ N(µi , 'x2), i = 1,…, n and yj ~ N(µy,
'y2), j = 1,…, m. Each sample has a sample mean given by
x=
1
n
n
(
xi and y =
i =1
1
m
m
(y
(2.1)
j
j =1
respectively, and sample standard deviations given by
n
sx =
©2000 CRC Press LLC
( ( xi # x )
i =1
n #1
m
(( yj #y )
2
and sy =
j =1
m #1
2
(2.2)
If the population variances are equal ('x = 'y = '), then the statistic
defined by
T=
(x # y ) # (µ x # µ y )
sp
1 1
+
n m
(2.3)
has a t-distribution with n + m – 2 degrees of freedom. sp2 is a pooled estimate
of the common variance ' defined as
sp2 =
(n # 1)sx2 (m # 1)sy2
n+m#2
(2.4)
Given the null hypothesis of no difference, the term µx – µy may be
removed without effect. It is conventional among users of these tests to set
a “significance level” discussed earlier. Table 2.2 shows 1% critical values
from a standard t-distribution. These are the values which the test statistic
must exceed for the two means to be declared different at the 1% significance
level for various degrees of freedom. So if, for example, n + m – 2 = 10, then
the data shows that, if the null hypothesis is true, there is, on average, only
a 1% theoretical chance that the test statistic would exceed 3.169 by random
measurement error alone. If the statistic did indeed exceed this value, then
the null hypothesis would be rejected at the 1% significance level.
Table 2.2 Table of Critical Values for a tTest at 1% Significance Level
Critical value for test
at 1% significance
Degrees of freedom
level
5
6
7
8
9
10
11
12
13
14
15
16
4.032
3.707
3.499
3.355
3.250
3.169
3.106
3.055
3.012
2.977
2.947
2.921
2.2.2.1.2 Welch’s modification to the Student’s t-test. Welch’s modification to Student’s t-test82 uses the same procedure except that the test
statistic is slightly different and the degrees of freedom are nonintegral.
©2000 CRC Press LLC
As mentioned earlier, the observed variance in a group of glass has two
components: the intrinsic variance of the glass fragments and the measurement error.
It seems likely that the measurement error for recovered fragments will
be higher than for the control. Recovered fragments may be dirtier and
smaller than the fragments freshly broken from the control. This leads to a
less distinct disappearance/reappearance of the glass as it is cooled and
heated in the oil. The resultant measurement of the RI is less precise than
that for a better glass sample. Conversely, for the measurement of the control
glass, there is ample glass to mount on a microscope slide, and there is a
large number of glass fragment edges to choose from. This usually leads to
better measurements on the control glass.
Experiments have been described (Jones and Smalldon, personal communication) where the “choice” of fragment by the operator was removed.
In these experiments the variance of the “control” group was increased. They
have postulated that this is because of an unconscious choice mechanism
that tends to give the same result for consecutive measurements by the
operator selecting edges with a similar “look.”
Not relevant to this discussion (which is centered on the use of GRIM I
and II) but fascinating is the experiment by K.W. Smalldon (personal communication, 1995) where music was played to subjects using the “clicking”
Mettler Hot-Stage. Those subjects listening to music had larger variances in
their measurements, suggesting that the clicking was interacting in some
way with their pressing of the button to record disappearance. Because
disappearance and reappearance are visual phenomena, this suggests that
the clicking was biasing the results toward a lower variance in some way,
perhaps by suggesting when the operator should press the button.
It must also be noted that surface fragments will be overrepresented in
the recovered group, and this may also lead to an increase in variability of
the recovered group (Jones, K.W. Smalldon, and Crowley, personal communication).
The control sample submitted is not always a good sample. It often
contains only one or a few small pieces of glass, and these cannot be expected
to represent the full range of variability present in the window or windscreen. For example, Locke and Hayes47 measured the RIs across the diagonal
of a windscreen. The variance across the whole windscreen was higher than
that at any one point.
Therefore, since it is unlikely that the control sample is always truly a
random sample, the standard deviation of the control sample will underestimate the actual standard deviation or variance of the window.
A credible line of argument can be made that a larger variance for
recovered glass must be expected. We have previously mentioned that tests
(Reference 37 and C.A. Coulson, J.M. Curran, and A.B. Gummer, personal
communication, 1995) suggest that there is an increase in the variation of
measurement with lower edge count, but that the mean remains approximately the same. Since using GRIM, it is easily possible to get 99, 99 edge
©2000 CRC Press LLC
counts for control fragments, whereas recovered fragments are typically less.
It follows that the variance of the recovered group must be larger.
Whatever the explanation, and there are many contenders, it is an empirical fact that the standard deviation of the recovered group is typically larger,
often 50% larger, than the control. Pinchin and Buckleton (personal communication) collected data from three Forensic Science Service (FSS) laboratories
and observed that the variance was, in fact, typically larger for the recovered
group. A plot of the standard deviation of the recovered groups against the
control for this data is given in Figure 2.4. It can be seen that most of the
points lie above the x = y line.
0.00018
Recovered std. deviation
0.00016
0.00014
0.00012
0.00010
0.00008
0.00006
0.00004
0.00002
0.00000
0.00000
0.00005
0.00010
0.00015
Control std. deviation
Figure 2.4 Standard deviation of recovered groups against controls for data from
R.A. Pinchin and J.S. Buckleton, 1993.
We have experienced situations where the control group fragments seem
very close in RI, giving a very small estimate for the standard deviation of
this group. One possible result of this is that the samples may fail the t-test
even though the groups are very close in absolute RI.
If the control and recovered groups are from different pieces of glass
then well and good, but if it is an artifact of the higher variability of the
recovered group and the low variability of the control group then the result
is unacceptable and a remedy is required. We point out that this is a minor
point in the context of glass examination and has been described as a mindless detail. It can easily and justifiably be argued that the t-test would outperform range tests by an even larger margin if the variance of the recovered
samples was high. We proceed here, however, to offer an option to those
glass examiners who are concerned about some of these details.
©2000 CRC Press LLC
Often the size of the recovered sample is small, possibly as small as 2
or 3 fragments, while the control sample can be arbitrarily large. Usual
practice takes a control sample of 6 to 10 fragments. In such cases the estimate
of the pooled standard deviation is dominated by the larger sample in the
t-test procedure and the larger variance of the recovered fragments goes
largely unnoticed.
To overcome the problem of false discrimination because of poor variance estimates, we advocate the use of Welch's modification82 to the t-test.
This makes allowance for the variance of the recovered group and the control
group to be unequal. This means that assumption A1(b) is not made; however, the remaining assumptions are.
Welch’s statistic is given by
V=
(x # y ) # (µ x # µ y )
sx2
n
+
sy2
(2.5)
m
Again, assumption A1(a) simplifies this expression to
V=
(x # y )
sx2
n
+
sy2
(2.6)
m
The distribution of V is no longer exactly t, but may be approximated
by a t) distribution whose degrees of freedom, ), may be estimated from the
data and are
)=
* s 2 sy2 x
, + /
+ n m.
*
sy4
sx4
+
, 2
/
2
+ n (n # 1) m (m # 1) .
(2.7)
which is typically nonintegral. This modified statistic is used as the default
to carry out two-sample t-tests in such widely available statistical packages
as SAS and exists as an option in Microsoft Excel.
Percentage points for t-distributions with nonintegral degrees of freedom can be quickly and accurately computed.83 If we take [) + 1], the integral
part of ) + 1, for the degrees of freedom, the percentage points can be
obtained from standard tables of the t-distribution. Using [) + 1] will provide
slightly smaller P-values than using the exact degrees of freedom, thus
©2000 CRC Press LLC
leading to conservative test procedures. Many studies have shown that the
performance of V in practice is entirely satisfactory.84,85
2.2.2.2 How many control fragments?
The practical consequences of examining the behavior of the testing procedures are that if the number of recovered samples is small, there is no benefit
in estimating the RI of a large number of control samples and there may be
some harm.
Since the size of the recovered sample, m, is often beyond the control of
the investigator, whereas the size of the control sample, n, can be made very
large, use of the t-statistic will invariably lead to domination of the estimate
of the standard error of the difference of the means by the standard deviation
of the control sample. This is untenable for two reasons. The first, discussed
earlier, is the underestimation of the standard error. The second arises from
the number of degrees of freedom (d.f.) of the t-test, n + m – 2, which will
apparently cause the test to become more sensitive as n increases.
The behavior of t and V in the limit where m, the size of the recovered
sample, is small and n, the size of the control sample, is very large is instructive.
T0
x#y
x#y
with n + m # 2 d.f. and V 0
with m # 1 d.f.
sx m
sy m
(2.8)
Using T to test the hypothesis that µx = µy is tantamount to testing
whether y— can be regarded as coming from a normal distribution with mean
—
—
x and a variance 'x2. On the other hand, the use of V tests whether y comes
—
from a normal distribution with mean x and a variance 'y2. Since we expect
that in practice 'y2 > 'x2, and since V will be tested against a t-distribution
with many fewer degrees of freedom, routine use of the t-statistic will lead
to oversensitive tests.
Even if Welch’s test is used, there appears to be a “law of diminishing
returns.” Increasing the number of control measurements should improve
the estimates of 'x2 and —
x. However, there is not an equivalent improvement
in the estimation of
sx2
n
+
sy2
m
(2.9)
because as n becomes large this is dominated by sy2/m which is fixed by the
recovered sample obtained. This suggests that the number of control fragments can be set relative to the number of recovered fragments obtained (see
Table 2.3).
©2000 CRC Press LLC
Table 2.3 A Plausible Number of Controls to
Take Conditional on the Size of the Recovered
Group
Number of recovered Recommended number
fragments (m)
of controls (n)
1
4
2
4
3
6
4
6
5
8
6
9
7
10
8
10
9
10
10
10
12
10
15
10
20
10
We have investigated the performance of these combinations relative to
the nominal ! level by simulation; however, we are still investigating the
logic of these simulations and do not present them here.
2.2.2.3 Setting significance levels
If we assume that a significance level is chosen, say !, this may be interpreted
as the theoretical false exclusion rate. This false exclusion rate is theoretical
in that it depends on a number of assumptions being met, which, to a greater
or lesser extent, may be violated as discussed previously. In such a case the
empirical false exclusion may be more or less than 1%. However, let us
proceed with the concept that the true false exclusion rate is !. It is necessary
to set ! in order to proceed with the concept of hypothesis testing.
A compelling argument might be to set ! such that the false exclusion
rate ! plus the false inclusion rate 1 is minimized. Since 1 is case dependent,
this would have to be performed on a per case basis. This is, nonetheless,
the logical conclusion of the hypothesis testing approach or, alternatively,
the first faltering steps to the Bayesian approach.
Some countries have a legal maxim, “it is better that 20 guilty men go
free than to convict one innocent man.” Can this be used? At face value this
might suggest that a viable procedure would be to set ! = 20 1. This holds
if, and only if, matching and nonmatching groups occur in equal proportions
(this follows from a chain of logic more prevalently discussed in the field of
DNA evidence called the fallacy of the transposed conditional or the prosecutors which we will explain later). At face value then, if we wish to set
the false exclusion rate to 20 times the false inclusion rate and if the probability of the group matching or not before measurement of the RIs is equal,
then we set ! = 20 1. This is not normal forensic practice.
©2000 CRC Press LLC
2.2.2.4 Elemental composition measurements — Hotelling’s T2
Researchers have found that elemental composition comparisons add discrimination potential to distinguish between glass fragments when the RI
does not.41,58,66 The elements of interest are the minor and trace elements:
aluminium (Al), iron (Fe), magnesium (Mg), manganese (Mn), strontium
(Sr), zirconium (Zr), calcium (Ca), barium (Ba), and titanium (Ti). Trace
elements such as strontium and zirconium are present in the low parts per
million concentration range, and preliminary work on small data sets has
shown that these elements have little or no observable correlation between
the other elements,41,58,66,86,87 which makes them very good discriminating
“probes.”
Traditional treatment of the data involves determining the mean concentration and the standard deviation for each element and then comparing
the means using a “3 sigma rule” and testing the match criteria to determine
if the ranges overlap for all of the elements. If any of the elements fails this
test, then the fragments are considered not to match.
The following section will demonstrate a statistical test that has advantages over the 3 sigma rule approach.
2.2.2.4.1 The multiple comparison problem. The 3 sigma rule has two
problems. The first is the problem of multiple comparisons. The 3 sigma rule
has a theoretical false rejection rate of approximately 0.1%, i.e., on average
one time in one thousand the scientist will say the mean concentrations are
different when, in fact, they are the same. This empirical false rejection rate
is much higher when small numbers of fragments are being compared. Each
comparison for each element has the same rate of false rejection; however,
the overall rate is much larger, even if the elemental concentrations are
independent. Consider the following: I have an extremely biased coin with
the probability of getting a tail equal to Pr(T) = p = 0.01. The outcome of
each coin toss is independent of any previous toss. If the number of tosses,
n, is fixed and X is the random variable that records the number of tails
observed, then X is binomially distributed with parameters n and p. This
experiment is analogous to making pairwise comparisons on n element
concentrations. If n = 10 and the probability of a false rejection on one element
is 0.01, then the probability that at least one false rejection will be made is
0.096 or 9.6%, so the overall false rejection rate is nearly ten times higher
than the desired rate. In general, if the false rejection rate, or size, of a
procedure is ! for a single comparison, and n comparisons are performed
in total, then the overall size is approximately n × !. A simple solution is to
increase the width of the intervals so that the size of the individual comparison is !/n. This is known as the Bonferroni correction and its immediate
drawback is obvious — as n, the number of elements, increases, it becomes
almost impossible to detect any difference between the two means for any
given element.
©2000 CRC Press LLC
The second problem with the 3 sigma rule approach is that is fails to
take into account any estimated correlation between the elements. That is,
the estimated concentration of one element will be associated with the estimated concentration of another element. Failure to include this information
results in severe underestimation of any joint probability calculation.
The solution to both these problems is the multivariate analog to the
Student’s t-test.
2.2.2.4.2 Hotelling’s T2 — A method for comparing two multivariate mean
vectors. The Student’s t-test and Welch’s modification have been used and
discussed extensively in the treatment of RI measurements.8,75 Comparison
of glass samples with respect to RIs examines the standardized distance
between the two sample means. Hotelling’s T2 (named after Harold Hotelling, the first statistician to obtain the distribution of the T2 statistic) is a
multivariate analog of the t-test that examines the standardized squared
distance between two points in p-dimensional space.88 These two points, of
course, are given by the estimated mean concentration of the discriminating
elements in both samples.
Suppose that m fragments have been recovered from the suspect, n
control fragments have been selected from a crime scene sample, and n + m
> p + 1, where p is the number of elements considered, then T2 has a scaled
F-distribution
T2 ~
( n + m # 2) p
(n + m # p # 1)
Fp ,n+ m# p#1
(2.10)
Use of the F-distribution depends on two assumptions about the statistical distribution of the data: (1) both samples come from a multivariate
normal distribution and (2) both populations have the same covariance
structure,89 i.e., the spread of the elemental concentration is approximately
the same in each. Large values of T2 are evidence against the hypothesis of
no difference between the two populations, i.e., evidence against a match.88
2.2.2.4.3 Examples. The data in the following examples come from
two distinct sources, one brown bottle and one colorless bottle taken from
a different process line at the same plant at the same time. Ten fragments
were taken from each bottle, and the concentrations of aluminum, calcium,
barium, iron, and magnesium (p = 5) were determined by ICP-AES.
The first example uses five fragments from the brown bottle as a control
sample (n = 5) and five fragments from the same bottle as a recovered sample
(m = 5) so that the population means are truly equal. Hotelling’s T2 = 11.69
and F5,4 (0.01) = 15.521, so
T 2 = 11.69 <<
©2000 CRC Press LLC
( 5 + 5 # 2 )5
(5 + 5 # 5 # 1)
F5 , 4 (0.01) = 10 F5 , 4 (0.01) = 155.21
(2.11)
T2 is comparatively small in relation to F, thus there is no evidence to reject
the null hypothesis, i.e., there is no evidence to suggest that the two samples
come from different sources.
The second example takes ten fragments from the brown bottle as the
control sample (n = 10) and ten fragments from the colorless bottle as the
recovered sample (m = 10), so the null hypothesis is false, i.e., the population
means are truly different. Hotelling’s T2 = 708.86 and F5,14 (0.01) = 4.69, so
T 2 = 708.86 >>
(10 + 10 # 2)5
(10 + 10 # 5 # 1)
F5 ,14 (0.01) =
90
F (0.01) = 30.15 (2.12)
14 5 ,14
In this example T2 is comparatively large in relation to F, thus there is very
strong evidence to reject the null hypothesis, i.e., there is very strong evidence to suggest that the two samples come from different sources.
2.2.2.4.4 Discussion on the use of Hotelling’s T2. Hotelling’s T2 test for
the difference in two mean vectors provides a valid statistical method for
the discrimination between two samples of glass based on elemental data.
The user may decide whether or not to include an RI measurement comparison, but the test remains the same. The properties of the test are closer to
the desired properties of the 3 sigma rule than the rule itself. Hotelling’s T2
must be advocated in place of any algorithms based on the 3 sigma rule or
modifications of it.
In general, however, it would be desirable to abandon tests altogether
and move toward a direct calculation of a likelihood ratio from continuous
multivariate data.
2.3 Coincidence probabilities
Consider a case in which there is one group of recovered fragments which
is indistinguishable from the control on the basis of the t-test at the 1% level
and on n + m – 2 degrees of freedom. A conclusion that “the recovered glass
could have come from the same source as the control” is inadequate. Indeed,
it is little more than a statement of the obvious. If we are to be balanced in
our approach then we must add “but it could also have come from some
other source.” In order to assist the deliberation of a court of law, it is
necessary that we give some indication of the strength of the evidence.
The traditional forensic approach to this kind of problem is to invoke
the idea of “coincidence.” If the recovered fragments have not come from
the scene window then the match is a coincidence. What is the chance of
such a coincidence occurring? This leads to the question “if I take control
fragments from some glass source other than the crime window, what is the
chance they would match the recovered, using the same comparison test?”
Intuition tells us that the smaller this chance is, the greater the evidential
value of the observed match. Evett and Lambert75 were among the first to
©2000 CRC Press LLC
specifically define a coincidence probability. “The coincidence probability …
is the probability that a set of m fragments taken at random from some source
in the population ... would be found to be similar .... with n control measurements of mean ….” It should be noted that the definition given by Evett
and Lambert and the one referred to as desirable later in this book are similar,
but not the same. The former refers to fragments that would match the
control, and the latter refers to fragments that would match the recovered.
This difference was elegantly pointed out by Stoney.90 Deeper reflection on
the question has followed, championed in the forensic community by Evett,
and is continued in Chapter 3.
Following a precise definition of what is meant by “coincidence probability,” it is easy to develop an algorithm that can estimate this probability.
We are unable, however, to give a precise (and correct) definition of
coincidence probability. Later, we will want the probability of this evidence
(the recovered fragments) if they have not come from the control. Since we
are assessing the probability of the evidence if the glass did not come from
the control window, it is difficult to see how the n measurements of the
control have anything to do with the definition.
However, in an attempt to proceed, we define this probability as an
answer to the following question. “What is the probability that a group of
m fragments taken from the clothing of a person also taken at random from
the population of persons unconnected with this crime would have a mean
RI within a match window of the recovered mean?” Defining the match
window is also difficult, and most attempts are based on applying the match
criterion to m fragments from random clothing (or other population) and
the n fragments from the control. Such a match window would be centered
on the recovered mean rather than the control mean, as implied in the
question given by Evett, but of the same width as that implied by Evett.
Common sense tells us that the smaller the coincidence probability is,
the greater the evidential value of the observed match.91 The coincidence
probability may be interpreted as the probability of false inclusion, 1, referred
to earlier.
Although this coincidence probability approach has strong intuitive
appeal, it has many problems. What do we mean by “some glass source
other than the crime window?” Do we mean another window? What other
kinds of glass sources should we consider?
The more we think about these questions, the more we are led to suspect
that our original question in relation to coincidence may not be the best one.
Our practices in this area have been conditioned largely by two factors. The
first is that it has always been easier to use as a source of data the control
glass samples that are submitted during routine casework. The second factor
is the availability of elemental analysis, which in many cases can establish
whether or not the recovered fragments are window glass.
Another problem arises when, for example, we have two groups of
recovered fragments — one matches the control and the other does not.
©2000 CRC Press LLC
When we try to formulate our coincidence probability we soon run into
logical inconsistencies which have never been fully resolved and which were
discussed in later work by Evett and Lambert.92
Whatever we do with the coincidence approach we must recognize that
we are taking a fairly narrow look at the evidence.93,94 Intuitively, we will
recognize that there are other considerations which may be every bit as
important, particularly those of transfer and persistence. Although research
has been done which may help us assess the likelihood of transfer and
persistence in a given case, we need some kind of logical framework to roll
that assessment into the overall appreciation of evidential strength. The
coincidence approach has no means of doing this.
2.4 Summary
The conventional approach to glass evidence described here is essentially a
two-stage approach: matching and coincidence estimation. As we have seen,
there are a number of problems.
1. The principles of this kind of approach have never been clearly
established, and the approach does not have a clear, logical framework.
2. The matching stage is open to challenge if it cannot be seen to embody
clear statistical principles. Yet significance tests offer an air of objectivity that is illusory because they inevitably require assumptions, the
validity of which in any particular case involves human judgment.
3. The choice of significance level for the matching test is entirely arbitrary, and the sharp distinction between match and nonmatch does
not make much sense.
4. The coincidence approach to estimating evidential weight does not
lead to the formulation of pertinent questions when considering the
defense perspective. In particular, it is incapable of dealing with problems of multiple controls and/or recovered groups.
5. The two-stage approach provides no framework for taking account
of the considerations of transfer and persistence in forming an overall
assessment of the evidence.
In order to interpret glass evidence, we require an infrastructure that
enables us to do the following.
1. Identify the questions which are not only relevant to the needs of the
criminal justice system, but are also within our capabilities to answer.
2. Work within a logical framework based on scientific principles.
3. Utilize the most useful statistical methods whenever they are relevant,
but subject to the disciplined use of expert judgment.
4. Design and create databases.
©2000 CRC Press LLC
5. Utilize the available databases effectively within the context of expert
judgment.
6. Identify and explore all of the hypotheses which might be relevant in
an individual case, giving them weight in a balanced and scientific
manner.
As we will see in the next chapter, using the Bayesian framework allows
us to satisfy most of these criteria.
2.5 Appendix A
Scott Knott Modification 2 (SKM2)
1. Sort y1, y2,…, ym into ascending order and label the sorted observations
y(1), y(2),…, y(m).
2. For j = 1 to m – 1, calculate the following
2
==*
*
B j = j y1 # y + ( k # j ) y 2 # y
+
.
+
.
2
(2.13)
where —
y1 is the mean of the fragments y(1) to y(j), —
y2 is the mean of
=
the fragments y(j+1) to y(m), and y is the mean of all the fragments, i.e.,
calculate the BGSS for each of the m – 1 ordered partitions of the data
into two subgroups.
3. Find the maximum BGSS B0 and record the position where it was
found (say j*).
4. Calculate the statistic proposed by Scott and Knott,80
%3 =
B0
2
2(2 # 2) s02
(2.14)
5. If %* > %* (!;m), then split the fragments y(1), y(2),…, y(m) into two groups,
{y(1), y(2),…, y(j*)} and {y(j*+1), y(2),…, y(m)}.
6. If there was a split in Step 4, repeat Steps 2 through 5 for each new
subgroup until no more splits can be made.
Evett Lambert 1 (EL1)
1. Sort y1, y2,…, ym into ascending order and label the sorted observations
y(1), y(2),…, y(m).
2. Select x(M), where M is the greatest integer less than or equal to 1/2 (m
+ 1), e.g., if m = 11, M = 6.
©2000 CRC Press LLC
3. Find the next nearest fragment to y(M), which will be y(M–1) or y(M+1),
and combine it with y(M) to form a group of 2. If the range of the group
is within the required limits, find the next nearest fragment and compare the range for a group of size 3, and so on. The critical values
r(!;m) are listed in Table 2.4 in Appendix B.
4. Repeat Step 3 until all the elements are in one group, or go to Step 2
for any subgroups of size >1.
Evett Lambert Modification 1 (ELM1)
1. Sort y1, y2,…, ym into ascending order and label the sorted observations
y(1), y(2),…, y(m).
2. Check that the range of y(1), y(2),…, y(m), r = y(m) – y(1) is less than or
equal to the 100(1–!)% critical value in Table 2.5 in Appendix B, r(!;m).
If r 4 r(!;m), then the fragments are considered to have come from
only one source, and, therefore, there is no need to go any further. If
this is not the case, i.e., r > r(!;m), then go to Step 3.
3. Select y(M), where M is the greatest integer less than or equal to 1/2
(m+ 1), e.g., m = 11, M = 6.
4. Find the next nearest fragment to y(M), which will be y(M–1) or y(M+1),
and combine it with y(M) to form a group of 2. If the range of the
group, r, is within the required limits, find the next nearest fragment
and compare the range for a group of size 3, and so on. The required
limits are listed in Table 2.5 in Appendix B.
5. Repeat Step 3 until all the elements are in one group, or go to Step 2
for any subgroups of size >1.
Evett and Lambert Modification 2 (ELM2)
1. Sort y1, y2,…, ym into ascending order and label the sorted observations
y(1), y(2),…, y(m).
2. Check that the range of y(1), y(2),…, y(m), r = y(m) – y(1) is less than or
equal to the 100(1 – !)% critical value in Table 2.6 in Appendix B,
r(!;m). If r 4 r(!;m), then the fragments are considered to have come
from only one source, and, therefore, there is no need to go any further.
If this is not the case, i.e., r > r(!;m), then go to Step 3.
3. Find the smallest gap between the fragments, i.e., find i and j such
that |y(i) – y(j)| is minimized for 1 4 j < i 4 m.
4. If |y(i) – y(j)|4 r(!;2), i.e., if the range of the group consisting of
fragments y(i) and y(j) is less than the range of a group of size 2 from
a normal distribution, find the nearest neighbor to y(i) or y(j) and
compare the range against r(!;3), etc.
5. Repeat Step 4 until all the elements are in one group, or go to Step 2
for any subgroups of size >1.
©2000 CRC Press LLC
2.6 Appendix B
Table 2.4
90, 95, and 99% Critical Values for %* Given m
%*(0.01;m)
3.73
5.74
7.39
8.88
10.23
11.47
12.81
13.83
15.18
16.42
17.42
18.79
19.78
20.89
22.13
22.91
24.25
25.42
26.45
m
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
%*(0.05;m)
5.33
7.38
9.56
10.91
12.32
13.61
15.23
16.37
17.54
18.89
19.84
1.24
22.49
23.77
24.80
25.53
27.03
28.31
29.20
%*(0.01;m)
9.29
11.81
13.46
15.21
16.94
18.29
20.64
21.82
23.09
24.20
25.44
27.03
28.02
29.84
31.40
31.61
32.66
34.05
35.45
Table 2.5 95% Critical Values for the Range of a
Sample of Size m from a Normal Distribution Scaled
by a Weighted Estimate of Standard Deviation75
m
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
©2000 CRC Press LLC
10,000 × 95th percentile
1.13
1.46
1.68
1.86
1.98
2.09
2.21
2.29
2.36
2.44
2.50
2.56
2.61
2.66
2.70
2.74
2.78
2.82
2.85
Table 2.6 90, 95, and 99% Critical Values for
the Range of a Sample of Size m from a
N(0,s2) Distribution with s = 4 × 10–5
m
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
90th
95th
percentile percentile
r(0.1;m)
r(0,05;m)
0.93
1.11
1.16
1.32
1.28
1.44
1.38
1.54
1.46
1.60
1.52
1.65
1.57
1.71
1.62
1.76
1.65
1.80
1.70
1.83
1.71
1.84
1.74
1.88
1.77
1.88
1.79
1.92
1.80
1.94
1.81
1.94
1.85
1.99
1.86
1.98
1.87
2.00
99th
percentile
r(0.01;m)
1.47
1.64
1.76
1.82
1.89
1.94
2.01
2.06
2.05
2.09
2.13
2.14
2.15
2.16
2.19
2.19
2.25
2.20
2.27
2.7 Appendix C88
Suppose that xi = [xi1,…, xip]T are the elemental concentrations of p elements
on the ith control fragment, and yj = [yj1,…, yjp]T are the elemental concentrations of p elements on the jth recovered fragment. If n control fragments
are to be compared with m recovered fragments, the matrices
5x11 … xnp 8 5 y11 … y mp 8
:
: 7
7
7x12 L xnp : 7 y12 K y mp :
:
:, 7
7
:
: 7
7
7M L M : 7M L M :
:
: 7
7
76x1p … xnp :9 76 y1p … y mp :9
(2.15)
represent the measurements. The summary statistics are defined by the mean
vectors
x=
1
n
n
(
xi and y =
i =1
and the sample covariance matrices.
©2000 CRC Press LLC
1
m
m
(y
j =1
j
(2.16)
SX =
1
n#1
n
((
j =1
)(
xj # x xj # x
)
T
and SY =
1
m#1
m
( (y # y )(y # y )
j
j
T
(2.17)
j =1
respectively. An estimate of the common covariance matrix, (, is given by
Spooled =
(n # 1)SX + (m # 1)SY
n+m#2
(2.18)
Hotelling’s T2 is then defined as
[
T 2 = ( x # y ) ( n1 +
©2000 CRC Press LLC
1
m
#1
)Spooled ] (x # y )T
(2.19)
chapter three
The Bayesian approach to
evidence interpretation
The most important task of the forensic scientist is to interpret evidence. As
we have seen, the conventional way to approach evidence does not allow
one to take into consideration parameters such as the presence of glass,
transfer, and persistence. Nor does it allow one to assess correctly the value
of glass when there are multiple control and recovered groups. Therefore,
several authors5-7,94-98 have indicated that Bayesian inference is the appropriate approach. Indeed, this approach allows the expert not only to address
the right questions to assist the court and elaborate research projects focused
on specific parameters, but it also allows the expert to avoid pitfalls of
intuition.
The application of this thinking to the interpretation of glass evidence
was anticipated in other forensic and legal fields at a very early stage.99
Robertson and Vignaux94 draw our attention to an interesting example.
Suppose a jury is required to determine whether or not a child has been
abused. The jury is told by an expert that the child rocks and that only 3%
of nonabused children rock. The jury might be inclined to conclude that
because the “coincidence” probability (the probability that the child rocks if
he/she is not abused) is small it is safe to infer that the alternative (the child
is abused) is true.
Clearly, the jurors’ views would change if they were then told that 3%
of abused children also rock. The evidence is clearly equally probable
whether the child was abused or not and, therefore, has no evidential value.
This small thought experiment leads quickly to the realization that it is
the ratio of the probability of the evidence under each of two (or more)
hypotheses that defines evidential value.
More formal mathematics also leads to this conclusion, and this result
was given by the Reverend Thomas Bayes in 1763.100 Before stating Bayes
Theorem, it is necessary to make some definitions.
©2000 CRC Press LLC
3.1 Probability — some definitions
• Random experiment — an experiment in which the set of possible
outcomes is known, but the outcome for a particular experiment is
unknown. For example, tossing a fair coin. The outcomes are heads
and tails, but exactly which will come up on any one throw is unknown.
• Outcome — the result of a random experiment. In the coin tossing
example the outcomes are heads or tails. If the experiment is observing the value of a card drawn out of a deck, the outcome may be any
one of {2, 3, 4, 5, 6, 7, 8, 9, 10, J, Q, K, A}.
• Sample space — the set of all possible outcomes. The sample space is
sometimes denoted by !.
• Event — a set of one or more outcomes. For example, one might ask
about the event of drawing a 10 or higher from a deck of cards. This
event, E, is defined by the set E = {10, J, Q, K, A}.
• Event occurring — an event is said to have “occurred” if the result of
a random experiment is one or more of the outcomes defined in the
event. For example, if a single card is drawn and it is a Jack, then the
event E given above is said to have occurred.
• Equiprobable or equally likely outcomes — outcomes of a random experiment are called “equiprobable” or “equally likely” if they have the
same chance of occurring.
Standard statistical texts often define the probability of an event two
ways, both closely related.
Definition
If all outcomes from a random experiment are equiprobable, then the
probability of an event E, Pr(E), is given by
Pr(E) =
# Outcomes in E
Total # of outcomes
(3.1)
Definition
The probability of event E is given by the long-term frequency that event E
occurs in a very large number of random experiments all conducted in the
same way.
These two definitions can be demonstrated by the coin tossing example.
Assume one is interested in the probability of a getting a head in a single
toss. If the coin is fair, then the probability of getting a head is equal to the
probability of getting a tail, i.e., the outcomes are equally likely. So if E = “a
head” = {H}, and the set of all outcomes is ! = {H,T}, then
©2000 CRC Press LLC
Pr( E ) =
# Outcomes in E
1
= = 0.5
Total # of outcomes 2
(3.2)
as we would expect. Similarly, we could repeat our experiment many times
and observe the frequency with which event E occurs, i.e., how many heads
do we see.
Table 3.1
A Coin Tossing Experiment
# Tosses
10
100
1000
10,000
100,000
# Heads in n tosses
6
52
463
4908
49,573
Frequency
0.60000
0.52000
0.46300
0.49080
0.49573
Table 3.1 shows a simulated coin tossing experiment. After 10 tosses, the
experimenters had obtained 6 heads, so they estimate the probability of a
head by the frequency, i.e., Pr(E) " 0.6. After 100 tosses, the experimenters
had seen 52 heads, so Pr(E) " 0.52. If the experimenters could continue this
experiment, i.e., for an infinite number of tosses, they would find Pr(E) = 0.5.
Most readers will be familiar with one or other of these frequentist or
classical definitions of probability. While these definitions form the basis of
modern statistics, they are unsatisfactory for the interpretation of scientific
evidence.
Definition
A probability is a rational measure of the degree of belief in the truth of an
assertion based on information.94
This second definition is a Bayesian or subjectivist definition of probability
and is one of the underlying concepts in the whole of Bayesian philosophy.
It should be apparent that this definition is more suited to our purposes. For
the purpose of illustration, take the question “what is the probability that
the suspect committed the crime?” There are two possible outcomes in the
“experiment” — either the suspect committed the crime or he did not.
However, we have no reason to believe that these outcomes are equally likely.
Similarly, the suspect may be a first-time offender, so a long-term frequency
approach will not work. The only way this question can be answered is with
the aid of evidence. We can also examine the coin tossing example in a
Bayesian framework. Realistically, we cannot toss the coin forever, and so
any long-term frequency is merely an approximation of the true probability.
At some point, however, our approximation will become sufficiently accurate
for us to believe that true probability of a head is a half. That is, the evidence
is sufficiently strong for us to form a belief about the probability of observing
a head. For a more thorough discussion on these concepts the reader is
advised to try either Robertson and Vignaux,94 Bernardo and Smith,101 or
Evett and Weir.102
©2000 CRC Press LLC
3.2 The laws of probability
There are three laws of probability. These laws hold regardless of whether
one subscribes to the frequentist or Bayesian definition of probability. We
define these laws for two events, but it is simple to generalize the results to
more than two events.
3.2.1
The first law of probability
The first law of probability is the simplest. It says that all probabilities lie
between zero and one. That is no probability can be less than zero or greater
than one. Mathematically, the first law of probability is
0 % Pr( A) % 1 for any event A
(3.3)
As a corollary to this, it is useful to note that the probability of any event
A that consists of no outcomes, i.e., the empty set, is zero (Pr(A = #) = 0). If
the event A consists of all the outcomes, i.e., the sample space, then the
probability of A is one (Pr(!) = 1). If A consists entirely of outcomes that are
not in the sample space, then the probability of A is zero.
3.2.2
The second law of probability
Definition
Two events A and B are said to be mutually exclusive if the occurrence of one
event precludes the other from happening.
For example, let A be the event that the coin comes up heads, and B be
the event that the coin comes up tails, then A and B are mutually exclusive
events.
Definition
The second law of probability: if A and B are mutually exclusive events, then
the probability of A or B occurring is
Pr( A or B) = Pr( A) + Pr(B)
(3.4)
Sometimes Pr(A or B) is written as the probability of A union B, Pr(A $ B).
Corollary
If A and B are mutually exhaustive, i.e., the outcomes in A and B make up the
set of all possible outcomes (A $ B = !), then Pr(A or B) = 1.
©2000 CRC Press LLC
Proof
Pr( A or B) = Pr( A $ B) = Pr(! ) = 1
If two (and only two) events A and B are mutually exhaustive, then A
–
and B are said to be complementary, and we write B as A.
3.2.3
The third law of probability
Before we can state the third law of probability, it is necessary to introduce
the concept of conditional probability. A conditional probability gives the
probability of a particular event conditional on the fact that some prior event,
say B, has occurred. This conditioning may have no effect at all or a very
large effect. We write this conditional probability as Pr(A|B) and read it as
the probability of A given B. All probabilities are conditional. For example,
in our coin tossing experiment, when we evaluated the probability of A given
B, we conditioned the probability on the fact that we knew the coin was fair.
Definition
The third law of probability: the joint probability of two events A and B is
given by
Pr( A and B) = Pr( A|B)Pr(B) = Pr(B| A)Pr( A) = Pr(B and A)
(3.5)
Two important concepts arise out of the third law of probability: those
of statistical independence and conditional independence.
Definition
Two events A and B are said to be statistically independent if, and only if,
Pr( A and B) = Pr( A)Pr(B)
(3.6)
Definition
Two events A and B are said to be conditionally independent of an event E if,
and only if,
Pr( A and B|E) = Pr( A|E)Pr(B|E)
3.2.4
(3.7)
The law of total probability
Finally, we use all three of these laws to define the law of total probability.
©2000 CRC Press LLC
Theorem
–
The law of total probability: if C and C are mutually exhaustive events, then
for some event E,
(
) ( )
Pr(E) = Pr(E|C )Pr(C) + Pr E|C Pr C
3.2.5
(3.8)
Bayes Theorem
Theorem
–
If C and C are mutually exhaustive events, then for some event E,
Pr(C|E) =
Pr(E|C)Pr(C)
(
) ( )
Pr(E|C)Pr(C) + Pr E|C Pr C
(3.9)
This definition is the traditional statistical definition, which does not
lend itself easily to description. However, using the laws of probability, it is
possible to restate Bayes Theorem using odds.
3.2.6
The relationship between probability and odds
We often hear about odds when we are talking about betting. In everyday
speech, odds and probability tend to be used interchangeably; this is a bad
practice because they are not the same at all.102
If we have some event C, then the odds in favor of C are given by
O(C) =
Pr(C)
( )
Pr C
=
Pr(C)
1 & Pr(C)
(3.10)
The odds on C are given by a ratio of two probabilities. This ratio can
range from 0 (when Pr(C) = 0) to infinity (when Pr(C) = 1). When the odds
are 1, they are referred to as evens. Let us examine our coin tossing example
once again. What are the odds on observing a head from a single toss? If C
is the event defined by “a head,” then as we have seen Pr(C) = 0.5. So the
odds on C are
O(C) =
0.5
0.5
=
=1
1 - 0.5 0.5
(3.11)
i.e., the odds are even. What about the odds on getting a card with a face
value of 10 or higher on a single draw from a fair deck (52 cards, no jokers)?
C = {10, J, Q, K, A}, so Pr(C) = 5/52, and the odds are
©2000 CRC Press LLC
O(C) =
Pr(C)
5 / 52
5
=
=
1 - Pr(C) 47 / 52 47
(3.12)
When the odds are less than one it is customary to invert them and call
them the odds against C.102 Here the odds are 47 to 5 against C. The concept
of odds can be extended to conditional odds. That is, for some event C and
some conditioning event E, the odds in favor of C given E are
O(C|E) =
Pr(C|E)
(
Pr C |E
)
Pr(C|E)
1 & Pr(C|E)
=
(3.13)
Once again, one can argue that all odds are conditional.
3.2.7
The odds form of Bayes Theorem
Theorem
–
If C and C are mutually exhaustive events, then for some event E,
Pr(C|E)
(
Pr C |E
)
=
Pr(E|C )
(
Pr E|C
)
×
Pr(C )
(3.14)
( )
Pr C
–
In the following proof it can be seen that the condition that C and C be
mutually exhaustive is not necessary for this form of Bayes Theorem. Several
authors have put forward arguments justifying the relaxation of this condition. However, these arguments have little or no effect on the overall result.
Proof
From the third law of probability
Pr(C and E ) = Pr(C | E )Pr( E ) and Pr(C and E ) = Pr(C | E )Pr( E )
(3.15)
and, conversely,
(
)
(
) ( )
(
)
(
)
Pr(E and C) = Pr(E|C)Pr(C) and Pr E and C = Pr E|C Pr C
(3.16)
But,
Pr(C and E) = Pr(E and C) and Pr C and E = Pr E and C
©2000 CRC Press LLC
(3.17)
so
(
)
(
) ( )
Pr(C|E)Pr(E) = Pr(E|C)Pr(C) and Pr C and E Pr(E) = Pr E|C Pr C
(3.18)
Now if we divide these two to find the odds on C given E, we get
Pr(C|E)
(
Pr C |E
)
=
Pr(E|C)
(
Pr E|C
)
×
Pr(C)
( )
Pr C
(3.19)
The odds form of Bayes Theorem lends itself to a very simple method
for updating prior belief. We may rewrite the expression 3.14 as
Posterior odds = Likelihood ratio × Prior
(3.20)
As it has been demonstrated by various authors,96,103 the evidence E alone
–
does not allow the expert to give his/her opinion on C or C . However, using
Bayes Theorem it is possible to show how the evidence influences the probabilities associated with the two competing hypotheses.
Therefore, the likelihood ratio (LR) converts the prior odds in favor of
C into the posterior odds in favor of C. It is this LR that the scientist will
attempt to estimate in order to evaluate the value of evidence. It has to be
noted that the background information I has been omitted for the sake of
brevity, but it must be remembered that the probability of the event can only
be measured in light of the circumstances of the case.* If the information I
is explicitly included, then
Pr(C | E, I ) Pr( E | C, I ) Pr(C | I )
.
=
×
Pr(C | E, I ) Pr( E | C , I ) Pr(C | I )
14243 144244
3 14243
posterior odds
LR
(3.21)
prior odds
Let us go back to the child abuse example. Let the event C be that the
child has been abused and the evidence E be that the child rocks. We are
told that 3% of abused children rock, i.e., that the probability that a child
will rock given he or she has been abused is 3%, Pr(E|C) = 0.03. If we are
–
told that 3% of children who have suffered no abuse rock as well, Pr(E|C )
= 0.03, then the odds form of Bayes Theorem tells us that
* One of the problems is that the suspect usually has no obligation to give any information or
that the information given may not be admissible; therefore, the information I may be provisional.
©2000 CRC Press LLC
Pr(E|C)
Posterior odds =
=
(
Pr E|C
)
× Prior odds
0.03
× Prior odds
0.03
(3.22)
= 1 × Prior odds
Posterior odds = Prior odds
We now have indisputable, mathematically justified proof that the evidence does not give support to either hypothesis. This simple but startling
revelation dispels all problems with coincidence thinking and also reveals
how all evidence may be viewed in a legal context. All that remains for this
chapter to show is how this philosophy may be applied.
3.3 Bayesian thinking in forensic glass analysis
In Evett’s first article specifically on the use of Bayesian approach to glass
evidence,6 Bayes Theorem was applied at the comparison stage. However,
as the method was not suitable for immediate application to casework, Evett
and Buckleton7 presented a pragmatic approach in order to benefit from the
Bayesian philosophy while awaiting ideal treatments.
The authors made three compromises. First, they assumed that there
existed a criterion in order to determine if the fragments did or did not match
(for example, Student’s t-test). Second, they assumed that the recovered
particles had been grouped before treatment. Third, it was also assumed that
there existed a means to estimate the frequency of occurrence of the observed
characteristics.
As the last approach is less complex, we will present first the work that
has been done on the use of Bayesian rationale after conventional tests and
then proceed with the continuous approach presented in Evett,6 Lindley,5 or
Aitken.96 In order to illustrate the interaction of the various aspects of glass
evidence, four different hypothetical cases will be presented: the first where
one control and one group have been recovered and the other cases where
multiple controls have been recovered.
Let us consider some of the thoughts that might occur to us before we
start the examination of the case. At this early stage it is worth attempting
to anticipate the opposing views that might be presented if the case did come
to court. The prosecution will present a simple view.
The suspect committed the offense.
The defense is not compelled to offer an alternative, but it may be
reasonable to suppose a sensible alternative, at least a priori, might be
©2000 CRC Press LLC
The suspect did not commit the offense.
It is necessary to approach our examinations with these opposing views
in mind. In doing so, we recognize that they are provisional and might
change with changing circumstances. We see here the definition of two
opposing hypotheses, a requirement for the Bayesian approach, and we
suggest one of the fundamental principles of evidence interpretation: evidence may only be interpreted in the context of two (or more*) hypotheses.
In 1997, Cook et al. (personal communication, 1997) discussed the hierarchy
of propositions that can be addressed in a case. These propositions fall into
three broad generic classes which are called source, activity, and offense
level. The offense level is generally the level the court addresses. The two
views of the defense and prosecution hypotheses at the offense level are
given in the previous example. It is customary for forensic scientists to avoid
such issues. Indeed, to address the offense level there is usually more than
one piece of evidence, and the judge or jury is more competent to weigh
both propositions. Hypotheses tested on the activity level can be addressed
by the criminalist. The propositions could be as follows:
C: the suspect smashed the window.
–
C : the suspect did not smash the window.
This level of the hierarchy presupposes that parameters such as transfer
and persistence can be estimated. If they cannot, then only the source level
can be addressed. The propositions of this last level are as follows:
C: the glass comes from the broken window.
–
C : the glass comes from another broken glass object.
When addressing the lowest level, it must be stressed that the value of
evidence will be either over- or underestimated. Therefore, in our opinion,
if nothing is known on the circumstances of the case, the expert should then
seriously consider not working the case.
Let us suppose that we address the activity level. We should, at this
stage, be asking ourselves what help we might be able to give in resolving
the conflict between these opposing hypotheses. The circumstances tell us
that if C is the case, then we can expect to find glass on the suspect’s clothing
which will match the control sample. How much glass?
•
•
•
•
No glass?
One or two small fragments?
A few small fragments?
Lots of small fragments?
* As there may be more than two hypotheses, Robertson and Vignaux94 suggested using
–
H1,H2,…, Hn instead of C and C .
©2000 CRC Press LLC
•
•
•
•
•
One or two big fragments?
A few big fragments?
Lots of big fragments?
A mixture of small and big fragments?
None of these?
–
What if C is the case? Bear in mind what we have been told about the
suspect.
Given the circumstances, what strength of evidence is likely to be
reported if C is actually the true hypothesis? If our expectation is that we
are only likely to find one or two small fragments, then we may not be able
to give more than a cautious opinion. If so, does the case examination
represent good value for money (Cook et al., personal communication, 1997).
–
What results are we likely to arrive at if C is actually the true hypothesis?
What if we had been told that the suspect has previous convictions for this
kind of offense? Whereas this would make him a good suspect from the
police perspective, it may well confound things with respect to evidence
transfer. If the suspect is a habitual law breaker, is it more or less likely that
we will find various kinds of glass on his clothing? Is he at greater risk of
being mistakenly associated with the scene than the average man in the
street?
Later, we will consider probabilities of transfer and persistence within
the context of this hypothetical case. However, the important point that we
want to make here is that we should think about such probabilities as much
before we find anything as after for two reasons: (1) because our expectations
should determine what, if anything, we are going to examine and how and
(2) because our post hoc assessments are likely to be colored by what we
have actually found. If one says “yes, that is just what I would have
expected,” it is not nearly as convincing as being able to demonstrate that
the findings were in accord with one’s predictions.
Suppose search of the clothing revealed a number of fragments of glass
on the surface. For the sake of generality we will not discuss, at this point,
any particular number, but rather state that n (= 1,2,3,...) fragments were
recovered from the suspect’s clothing. Similarly, let us avoid, for the time
being, the statistics of the measurements and take it that there was a clear
match between them and the measurements on the control.
The Bayesian (compromise) approach7 proceeds as follows.
Case 3.3.1 — One group, one control
Based on the eyewitness evidence and what is known about the suspect, we
visualize the prior odds in favor of C:
Prior odds =
©2000 CRC Press LLC
P(C | I )
P(C | I )
(3.23)
where I denotes all the nonscientific evidence which the court will take into
account. Let E denote the event that we have found n fragments on the
surface of the suspect’s clothing which match the control. Then we are
interested in the new posterior odds:
Posterior odds =
P(C|E, I )
(
P C |E, I
(3.24)
)
That is, the odds given that there is some evidence, E, is known in
–
addition to I which may lend support to C or C . This is where we use Bayes’
Theorem, which tells us that
P(C|E, I )
(
P C |E, I
)
=
P(E|C , I )
(
P E|C , I
)
×
P(C| I )
(
P C |I
)
(3.25)
This expression identifies a crucial factor, the LR:
LR =
P(E|C , I )
(
P E|C , I
)
(3.26)
To evaluate the LR we must answer two questions:
What is the probability of the evidence given that the prosecution hypothesis is correct and given the background information?
What is the probability of the evidence given that the defense
hypothesis is correct and given the background information?
These form the numerator and denominator of the LR. We will look at
the denominator first. The question, in more detail, is
What is the probability that we would find a single group
of n matching fragments on the surface of the suspect’s
clothing if he had not smashed the window at the scene,
given what we know about the incident and the suspect?
One problem, of course, is that we have no information about the suspect’s activities in the period leading up to the arrest, nor is he under any
obligation to provide such details in many jurisdictions. Furthermore, we
cannot be sure that any of the information given to us will be admissible;
but we have to start somewhere, recognizing that our adopted I is provisional.
©2000 CRC Press LLC
The way in which we proceed will be determined not only by the
circumstances of the case, but also by the expertise, knowledge, and data
that we have available to us. For the rest of this section we will base our
analysis on the Lambert, Satterthwaite, and Harrison (LSH) clothing survey.104 This survey will be discussed in more detail in Chapter 4. However,
at this stage we need to know that this survey can be used to give estimates
of nonmatching (random or background) glass on persons suspected of a
crime. Regardless of what we know about the suspect, it is inarguable that
he/she has come to police attention in connection with a breaking offense.
The casework clothing survey is a survey of people who also have come to
police notice in that way. This is a persuasive argument for considering this
survey to be the most relevant in this case. Let us see how we might use the
results. Let us break down E as follows:
1. One group of fragments was found on the surface of the suspect’s
clothing.
2. The group contained n fragments.
3. The group matched the control.
Using this breakdown, let us rephrase our question in relation to the
denominator:
If we examine the clothing of a man who has come to police
notice on suspicion of a breaking offense, yet he is unconnected with the offense, what is the probability that we will
find a single group of n fragments which match the control
in this particular case?
The LSH survey distinguished between groups of glass that matched
casework controls and those that did not. If we assume that the matching
fragments in each of these cases did, in fact, come from the incident being
investigated, then the nonmatching glass gives us a picture of the background glass that we can expect to see in such cases, when the suspect is
unconnected with the crime being investigated.
• Let Pi denote the probability that an innocent police suspect will have
i (i = 0, 1, 2,...) groups of glass on the surface of his clothing.
• Let Sj denote the probability that a group of glass fragments on the
surface of such a person’s clothing will consist of j fragments (j = 1,
2,…, n)
• Let f denote the probability that a group of glass fragments on the
surface of such a person’s clothing will match in RI the control in this
particular case.
To help in our analysis we make two further assumptions.
©2000 CRC Press LLC
A4: there is no association between the number of groups of glass found
on a person’s clothing and the sizes of those groups.
A5: there is no association between the frequency of a given RI of glass
on clothing with either the number of groups or the size of the group.
It is unlikely that either of these assumptions is exactly true; however,
we believe they are at least as correct as any first order approximation.
Experimental data suggest that there are no obvious strong correlations. We
note, defensively, that it is not that this approach requires more assumptions
than others, but rather it makes it easier for us to be clear about what the
assumptions are.
By invoking A4 and A5 and using our new terminology, we are now
able to answer our question in a logical fashion. That is, what is the probability that we will find a single group of n fragments which match the control
sample in this particular case on a person unconnected with the crime?
P( E | C , I ) = P1.Sn . f
(3.27)
Using the new data collection, we are able to address a question in
relation to the denominator, which is relevant to our original hypothesis,
–
C . This means that we can now turn our attention to the original C and
address the numerator P(E|C, I). The question, in more detail, is
What is the probability that we would find n matching
fragments on the surface of the suspect’s clothing if he had
smashed the window at the scene, given what we know about
the incident and the suspect?
The numerator is complicated slightly by the need to allow for at least*
two possible explanations for the evidence if C is the case:
1. Either the group of fragments was transferred from the scene window
— in which case, the suspect could not have had any other glass on
his clothing before.
2. Or no glass was transferred from the scene window, but the suspect
already had the group of glass fragments on his clothing.
Let Tk denote the probability that, given C and I, k (k = 0, 1, 2,…, n) glass
fragments would be found from the scene window on the suspect’s clothing.
Then, again invoking A4 and A5,
* Actually, there are n + 1 explanations: r fragments were transferred, n – r being there beforehand; r = 0, 1, 2,…, n. But it is possible to show that most of the terms associated with these
alternatives are small and leaving them out is, in any case, conservative, in the sense that we
are making the numerator smaller than it should be.
©2000 CRC Press LLC
P(E|C , I ) = T0 .P1 .Sn . f + Tn .P0
(3.28)
The first combination of terms deals with the second of the two alternatives: no glass being transferred, T0; one group of n fragments being there
beforehand, P1.Sn; which matched the control f. The second deals with the
other alternative: n fragments were transferred and none were there beforehand; we assume, for the time being, that if glass fragments were transferred
from the scene then they would match with probability 1. This is an approximation, but not a bad one.
We now have both the numerator and the denominator terms. Therefore,
the LR is
P(E|C , I )
(
P E|C , I
)
= T0 +
P0 .Tn
P1 .Sn . f
(3.29)
Case 3.3.2 — Two recovered and one control groups
The crime, arrest, and exhibits are the same as in Case 3.3.1. The only
difference is that ten fragments were found on the clothing surface. Based
on RI, these formed two clear groups: one of six fragments that matched the
control, and another of four fragments that was different from the control.
Again, our evaluation centers on the LR:
LR =
P(E|C , I )
(
P E|C , I
)
(3.30)
–
Once again, we will look at the denominator P(E| C, I) first and will
use the LSH data. The question now is
If we examine the clothing of a man who has come to police
notice on suspicion of a breaking offense, yet he is unconnected with the offense, what is the probability that we will
find two groups of fragments of the observed sizes and properties?
Let f1 and f2 denote the probabilities that groups of glass found on
clothing will have the RIs of the two observed groups. Let group 1 be the
matching group and group 2 be the nonmatching group. Then,
• P2 is the probability that the suspect’s clothing would have two groups
of glass.
• S6 f1 is the probability that a group of glass fragments on clothing will
have six fragments and the observed RI of group 1.
©2000 CRC Press LLC
• S4 f2 is the probability that a group of glass will have four fragments
and the observed RI of group 2.
Again, we are using assumptions A4 and A5, leading to
(
)
P E|C , I = 2 P2 S6 S4 f1 f2
(3.31)
The factor 2 arises because if we sample two groups of glass from clothing
there are two ways in which we can meet our requirements: the first group
can be size 6 and the second group size 4, or the first group can be size 4
and the second group size 6.
As in the first case, when we consider the numerator we have to allow
for two possibilities:
1. Either no glass was transferred from the scene window, but the suspect
already had the two groups of glass fragments on his clothing.
2. Or the group of six fragments was transferred from the scene window
— in which case, the suspect had a single group of four fragments on
his clothing beforehand.
Then,
P(E|C , I ) = T0 .2 P2 .S6 .S4 . f1 . f2 + T6 .P1 .S4 . f2
(3.32)
So the LR is
LR =
P(E|C , I )
(
P E|C , I
)
= T0 +
P1 .T6
2 P2 .S6 . f1
(3.33)
Unlike the conventional approach using the coincidence concept,92 the
frequency of the nonmatching group does not appear in the final evaluation.
Moreover, the value of the evidence is different if the groups are constituted
of six matching and four nonmatching fragments or four matching and six
nonmatching fragments. One can also see that the size of the nonmatching
group is irrelevant.
Case 3.3.3 — One recovered and two control groups
The scenario is the same as before, but two windows have been broken and
only one group of glass has been recovered on the suspect’s clothing. This
recovered glass matches control group 1. In order to estimate the probability
of coincidence, Evett and Lambert92 suggested summing the coincidence
©2000 CRC Press LLC
probabilities estimated for each control, so the value of the evidence
remained the same whether the recovered glass matched the more or the
less frequent glass. This problem does not arise in the Bayesian framework.
If the suspect was present when the window was broken, there are two
possibilities.
1. Sample 1 was transferred and recovered, but not Sample 2.
2. The glass is present at random, neither Sample 1 or 2 has been transferred from the scene windows.
If the suspect was not present when the window broke, then the glass
is present at random.
Assuming that
• T1k: The probability that k fragments have been transferred from control 1 have persisted and have been recovered.
• T2l: The probability that l fragments have been transferred from control 2 have persisted and have been recovered.
• Pi: The probability of finding i groups at random on the suspect’s
clothing
• Sn f1: The probability that a group of glass fragments on clothing will
have n fragments and the observed characteristics of group 1.
Then the LR is
LR =
P(E|C , I )
(
P E|C , I
)
=
T10 T20 P1Sn f1 + T1k T20 P0
P1Sn f1
= T10 T20 +
T1k T20 P0
P1Sn f1
(3.34)
If the probability of transfer is the same for controls 1 and 2, then
LR = T02 +
Tk T0 P0
P1Sn f1
(3.35)
Intuitively one would think that control 2 has to be taken into account;
indeed, the more controls the more probable it seems to have a match due
to chance. However, it is not the frequency of the intrinsic characteristics (RI
or other measurements on control 2) that influences the value of evidence,
but the fact that no glass has been recovered! It is T20 (in this case T0) that
diminishes the LR. Note also that unlike the coincidence approach,92 the LR
changes with the frequency of the intrinsic characteristics of the matching
group; if the recovered glass is rare, the LR is higher than if the recovered
glass is common.
©2000 CRC Press LLC
Case 3.3.4 — Two recovered and two control groups
The crime, arrest and circumstances are the same as in the case presented
before, but two groups of glass fragments have been recovered on the suspect’s garments. These two groups match the two controls. If we consider
the denominator, the two groups are present at random. For the numerator,
we have to allow for four possibilities.
1. The two groups of glass were transferred from scene windows one
and two, and the suspect had no glass on his clothing beforehand.
2. One group of glass came from scene window 1, no glass was transferred from scene window 2, but the suspect already had one group
of glass on his clothing.
3. One group of glass came from scene window 2, no glass was transferred from scene window 1, but the suspect already had one group
of glass on his clothing.
4. No glass was transferred from the two scene windows, but the suspect
already had the two groups of glass on his clothing beforehand.
So the LR is
LR =
LR =
P(E|C , I )
(
P E|C , I
)
=
T1k T2k P0
2 P2 SnSm f1 f2
T1k T2l P0 + T10 T2l P1Sn f1 + T1k T20 P1Sm f2 + T10 T20 .2 P2SnSm f1 f2
2 P2 SnSm f1 f2
+
T10 T2l P1
2 P2 Sm f2
+
T1k T20 P1
2 P2 Sn f1
(3.36)
+ T10 T20
If the probability of transfer is the same for controls 1 and 2, then
LR =
Tk2 P0
TTP
TTP
+ 0 k 1 + k 0 1 + T02
2
2 P2 Sn f1 f2 2 P2 Sn f2 2 P2 Sn f1
(3.37)
where Sm f2 is the probability that a group of glass fragments on clothing will
have n fragments and the observed characteristics of group 1.
The Bayesian approach, presented earlier in the different cases, shows
which parameters are important to consider when evaluating the value of a
match. It allows us also to explain what is the correct population to consider,
for example, why one should take into consideration the frequency of intrinsic characteristics of glass found at random and not of control glass. The
different scenarios demonstrate that even if there are several controls and
recovered glass and/or when there are nonmatching groups, the Bayesian
framework gives logical answers unlike the conventional approach.
©2000 CRC Press LLC
3.3.1 A generalized Bayesian formula
One of these terms typically predominates in cases with matching glass.
Therefore, it is possible to give a simple approximate general formula:
LR =
(G & M )! PG& M Ts1 Ts2 KTsm T0N
G! PG Ss1 Ss2 K Ssm f1 f2 K fm
(3.38)
where
• G is the total number of groups of glass on the clothing.
• M is the number of matching groups.
• N is the number of controls not apparently transferred. (A different
T0 is realistically expected for each control if the circumstances of each
breakage differ in any way. If such an extension is desired, the term
T0N should be replaced by the product of N T0 terms.)
• Tsi is the transfer probability for the group i of size Si.
• Si is the probability of a group of size si.
• fi is the frequency of the ith group.
This formula may be easily programmed into Microsoft Excel.
3.4 Taking account of further analyses
So far, the data we have considered consists solely of RI measurements. In
practice, we can often gather more data on both recovered and control
fragments. The methods of annealing and elemental analysis are widely used
in this regard. How do we take account of this additional data? Essentially,
there are two ways in which the added information can change our assessment. The first is that we may gain added discrimination from this information (assuming that a match results).
Recall that we defined f as “the probability that a group of glass fragments on the surface of a suspect’s clothing will match in RI in this particular
case.” We could, for example, redefine f as “the probability that a group of
glass fragments on the surface of a suspect’s clothing will match in RI and
elemental analysis in this particular case.” Assuming that elemental analysis
does indeed result in added discrimination, then the new f will be much
smaller than the old f and the LR will be bigger. However, in order to estimate
f, we must have a clothing survey that includes the results of elemental
analysis, requiring equipment which, at present, most laboratories do not
possess.
The second way in which additional information can affect our conclusions is potentially much more complicated. The additional data can affect
the hypotheses we consider. In particular, the new data can affect the way
©2000 CRC Press LLC
in which we will react to future provisional hypotheses that may be put
forward in court. The obvious example is where defense presents, possibly
as mere speculation, an alternative explanation for the presence of matching
glass. The most obvious explanation is that the defendant had recently been
in the vicinity of some other breaking glass object. If elemental analysis
enables us to classify the recovered fragments, then the probability of the
evidence under alternative hypotheses may be reduced or enhanced.
Imagine that the suspect says that he broke a beer jug recently. If either
interferometry or elemental analysis demonstrates that one or more of the
recovered fragments are from a flat (or flat float) source, then the probability
of this evidence, if the control glass is from a beer jug, is reduced (to zero if
we accept that the results are always correct and that all the recovered
fragments have come from one source). What if the elemental or interferometry analysis suggests that at least one fragment may have come from a
container source? It may be tempting to consider this fragment “eliminated”
and to proceed with assessing the remaining fragments in some way that
supports the prosecution hypothesis. We cannot support this approach. The
presence of even one confirmed container fragment in the group at this RI
greatly increases the support for the alternative hypothesis. We will develop
these hypotheses at more length later.
3.5 Search strategy
Extending Bayesian arguments suggests a logical method for search strategies and stopping rules. Consider the simplest formula given here: for one
control and one matching recovered group
P(E|C , I )
(
P E|C , I
)
= T0 +
P0 .Tn
P1 .Sn . f
(3.39)
P0
This formula is large whenever the ratio
is large. The terms in this
P
1 .Sn
formula relate to
• P0: the probability of finding no glass on the clothing of a random
person following THIS search strategy.
• P1Sn: the probability of finding one group of glass of size n on the
clothing of a random person following THIS search strategy.
Therefore, this ratio is most likely to be maximized by the following
strategy. The order of the examination should be
©2000 CRC Press LLC
1.
2.
3.
4.
5.
Hair combings
The surface of clothing
The surface (the uppers) of shoes
Pockets and cuffs of clothing
Glass embedded in shoes
This order is derived from an examination of Table 2 in both Harrison
et al.105 and Lambert et al.104
Searching should be stopped at some point. This point should be when
“most” of the evidence has been established. Chapter 4 will show that groups
of glass of size 3 or over are rare on persons unconnected to a crime. This
suggests that searching should be stopped when more than three fragments
have been found, and stopping when ten or more pieces of glass in total
have been recovered is certainly justified, especially if the next search would
be to move to a lower level in the hierarchy given previously.
It could be argued with some justification that this policy is designed to
optimize evidence from the prosecution point of view and, therefore, is
biased against the defendant. However, this same search strategy avoids
those areas most likely to produce “random” glass and, hence, to some extent
safeguards an innocent suspect against the largest risks of coincidental
matching.
No search strategy can safeguard an innocent suspect who coincidentally
has large amounts of matching glass on his/her hair or upper clothing.
This policy suggests that if, say, 20 fragments are found on the T-shirt,
the first garment searched, then about 10 of these should be examined. At
this point the control may be opened and the number of fragments recommended in Chapter 2 should be examined from the control (ten control
fragments if there are ten measured recovered fragments). If these “match,”
then the examination should be stopped. We seriously recommend stopping
without opening the lower clothing or shoes in such a case. This optimizes
evidence and saves time.
In the event that a high probability of transfer is expected and no glass
is found on the T-shirt, we also recommend stopping and writing a statement
supporting the suggestion that the clothing was NOT close to the window
when it was broken. The control should be examined after the search in case
there is evidence (such as the control is wired or laminated) that suggests
that much effort would have to be exerted to penetrate it, thereby increasing
the belief in high transfer probabilities.
We suggest that only in special circumstances should a search continue
all the way to the soles of the shoes (however, this matter should also be
discussed with the judge or police). The only circumstance that comes to
mind is that the window was broken by throwing an object from a distance
and the only suspected contact was the offender walking (gently) over the
glass.
©2000 CRC Press LLC
3.6 Comparison of measurements: the continuous
approach
In this section we introduce a method known as the continuous approach.
Under match/nonmatch thinking (as with Student’s t-test), the evidence is
a match until some predetermined point where it suddenly becomes a nonmatch. Smalldon (personal communication, 1995) termed this “the fall off
the cliff effect.” Under this thinking, a “good” match is assessed as having
the same evidential value as a “poor” match. A very narrow mismatch is
reported in the same way as an obvious mismatch. It is possible that the
glass did not pass a reasonable statistical test, but the RIs are still quite close
and there is the fact that there may be a lot of glass on the clothing.
The faults lie in the match/nonmatch approach and the sequential way
in which evidence is often considered. The solution lies in abandoning match
thinking, as suggested by Lindley.5
Abandoning the match/nonmatch approach is one of the great advances
of inferential thinking in forensic science. It has several advantages. Philosophically, it means that the forensic scientist is permitted to weigh all the
evidence in one process rather than in some step-by-step fashion.
We perform here a thought experiment. Imagine some comparison process between a control and recovered sample of something. Scientist 1 performs test 1 first. At test 1 the two samples narrowly mismatch using some
rule that has an error rate '. Most scientists are obliged to stop here and
report a mismatch, and indeed scientist 1 does so.
Scientist 2 starts at test 2 for some reason and finds a correlation of
features that are otherwise very rare in the population. She then performs
test 3 and finds another correlation of features that is otherwise rare. Tests
4, 5, and 6 also find a correlation of features that is otherwise very rare in
the population. Scientist 2 then finally performs test 1 and finds the narrow
mismatch. Such a scientist is presented typically with a very difficult problem. Most of the evidence suggests an association; however, one test has
found a mismatch. This test, however, is known to have a small error rate.
Has an error occured here? The match/mismatch approach is unable to
recover from this situation. In fact, the more tests that are performed on truly
matching samples, the more likely such an ambiguity will result by a false
exclusion occurring in one of the tests. How can this be? Surely more testing
must be better? The fault is in the match/nonmatch approach. This approach
may also be compared to the logic of Karl Popper. In this logical system a
hypothesis is advanced and subsequent tests attempt to disprove it. The
more tests the hypothesis survives, the more credible the hypothesis. The
forensic corollary of this hypothesis is that the two samples have the same
source. This is the logical system many of us were taught. However, return
to scientist 2. We suspect that her true belief is that the evidence supports
an association and that the close mismatch is an error in this case. She may
be obliged to report a mismatch or she may (indeed, in single locus work in
©2000 CRC Press LLC
6.9
6.7
6.5
6.3
6.1
5.9
5.7
5.5
5.3
5.1
4.9
4.7
4.5
DNA in the U.S. she might) report the close mismatch as inconclusive and
proceed to interpret the remaining tests. Neither answer is correct. The
correct approach lies, as we have suggested, in abandoning the match/nonmatch approach.
How can this be done? The method is no surprise to professional statisticians, but typically seems completely unfamiliar to forensic scientists.
Therefore, we spend some time introducing it.
Imagine another thought experiment. Scientist 3 has the same problem,
except that she is equipped with the scales of justice and a blindfold. Every
time she interprets a test, she puts the correct weight on the scales either on
the defense side or the prosecution side as is warranted. For test 1, the narrow
mismatch, she puts the correct weight on the defense side. For tests 2 to 6
she puts the weights on the prosecution side. In the end she simply reports
the resulting net weight and which side it supports.
If we have convinced anyone that this latter approach is highly desirable,
then all that is left is to show how to implement it. The implementation of
this approach is standard statistical methodology.
The basis of the implementation is the use of probability density rather
than probability itself. The concept of a probability density is novel to many
forensic scientists and seems very difficult to explain. One reason for this is
that it is often not taught at school, and indeed when it is taught, typically,
it is mistaught.
Most of us have been shown a normal distribution at school, possibly
something like Figure 3.1. This is intended to be a distribution of men’s
heights. It peaks somewhere just below 6 ft. The next question is simple since
we are all so familiar with this graph. What is the y-axis?
Height (ft)
Figure 3.1
A distribution of men’s heights.
We cannot tell what you answered or whether you realized that you
have probably never been told or shown this. If you answered probability
then you were wrong. The areas under this graph are the probabilities.
Perhaps you answered counts or relative frequency, both of these are also
©2000 CRC Press LLC
6.9
6.7
6.5
6.3
6.1
5.9
5.5
Men
5.3
5.1
4.9
4.7
4.5
Women
5.7
Probability density
0.14
0.12
0.1
0.08
0.06
0.04
0.02
0
Height (ft)
Figure 3.2
A distribution of men’s and women’s heights.
wrong (although relative frequency might be close). The y-axis is probability
density. This tells us that there is more probability about the mean and less
in the tails. It is not a concept that comes easily, but it is relatively easy to
use. Another graph is shown in Figure 3.2.
The white line is intended to be a survey of women’s height superimposed on the previous one of men’s heights. We can ask various questions
about this graph, some of which are easy to answer and some of which are
not. At the risk of spreading confusion, we initially ask “what is the probability that a man is 6.3 ft?” This question is unanswerable. All that can be
answered is how many men are between 6.25 and 6.35 ft (or a similar
question). We must define a width to get an area on this graph. Once we
define the width, we can answer the question. In this case the answer is
0.067444, which we have obtained by taking the area under the men’s graph
between 6.25 and 6.35 ft. Has this got anything to do with forensic science?
What is the probability that a set of recovered fragments will have an RI of
1.5168? First, you must tell me the width you want (the match window
perhaps?) then I can answer the question.
The need to define a width to calculate a probability leads inevitably to
the need to define a match window and to troubles associated with that.
As another thought experiment let us try the following. I have a person
who is 6.3 ft tall. Do you think it is a man or a woman? The evidence supports
it being a man. Why? Because the men’s graph is higher at the 6.3 ft mark
than the women’s. In fact, the exact support for this suggestion is the ratio
of the heights of the two graphs at this point. The ratio in this case is 11.3.
Can this concept be used in any way? It has the potential to avoid all
questions about matching and match windows. It has the potential to weigh
all evidence in the way suggested in the scientist 3 thought experiment. It
has the potential to weigh close matches, poorer matches, and mismatches.
This is done by constructing probability density distributions for glass evidence.
©2000 CRC Press LLC
As a practical step in the implementation we make some simplifications.
We propose to retain the grouping assumption for the benefits of simplicity
that follow from it, but to drop the match/nonmatch approach. The future
may involve dropping the grouping approach as well, but at this point we
cannot implement that suggestion.
As a simple way of imagining this transition, recall the formula for the
example given earlier for n matching fragments and no mismatching glass.
P(E|C , I )
(
P E|C , I
)
= T0 +
P0 .Tn
P1 .Sn . f
(3.40)
This, we have previously suggested, can be approximated by
P(E|C , I )
(
P E|C , I
)
=
P0 .Tn
P1 .Sn . f
(3.41)
There is a 1/f term in this equation. This is the probability of a match if
they are from the same source (1) divided by the probability of a match if
they are not (f). It is this 1/f term that can be replaced by the respective
probability densities. The numerator (previously 1) becomes a term relating
to the closeness of the match and the denominator (previously f) becomes a
term relating to the rareness of the glass. It can be seen that there is a very
clear correspondence between the two approaches.
The formula in this case (to replace the 1/f term) is
(
P x & y |x , Sx , Sy , C
(
P y |x , Sx , Sy , C
)
)
(3.42)
which can be interpreted as a numerator expressing the probability density
that the two means of the recovered and control are this close, over a denominator of the probability density that the recovered mean is as observed.
In Appendix A, we give our justification for this formula. It will be noted
that we have omitted for simplicity small positive terms such as T0. The
omission of these small positive terms has a negligible effect in most cases,
but in any case this effect is to the advantage of the defendant.
We return to Case 3.1.1 which we reinterpret using the continuous
approach. An excerpt of the probability density function of the RI of glass
on clothing from the LSH survey is given in Figure 3.3. The probability
density function of the difference of the sample means on the same scale is
also given in Figure 3.3.
This distribution is used in Welch’s modification of Student’s t-test and
assumes that the variances of the populations from which the two samples
©2000 CRC Press LLC
1.51826
1.51816
1.51806
Welch Density
LSH Density
1.51796
Density
5000
4500
4000
3500
3000
2500
2000
1500
1000
500
0
RI
Figure 3.3
Welch and LSH densities in the area about RI = 1.5180.
are drawn are unknown and unequal. The justification for this appears in
the appendix.* At the recovered mean 1.51804, the values are 4713 for the
Welch density and 120 for the LSH density (see Figure 3.3). It can be seen
that the LSH density is essentially a low horizontal line on this scale.
We now define:
where
• f(y– ) is the value of the probability density for glass at the mean of the
recovered sample (from Figure 3.3).
• p(x– – y– |s x s y) is the value of the probability density for the difference
between the sample means (from Figure 3.3).
As before, we take approximate values for the P and S terms from LSH
LR = 0.079 +
0.25 × 0.042 × 4713
0.22 × 0.02 × 120
(3.43)
= 94
which compares with the value 60 previously obtained and reflects the close
nature of the match.
* Strictly we wish to model the term p(X, Y|C). The vectors X and Y are the observations from
the control and recovered groups. This is, however, very complex, and solutions to date involve
a large number of assumptions. Modeling the more assessable term given is not likely to differ
in a meaningful way.
©2000 CRC Press LLC
3.6.1 A continuous LR approach to the interpretation of elemental
composition measurements from forensic glass evidence
In Chapter 2, Hotelling’s T2 test, a multivariate equivalent of Student’s t-test,
for determining a match between glass fragments recovered from a suspect
and a control sample of glass fragments was introduced. While Hotelling’s
T2 test is certainly a better approach than the 3 sigma rule, it is still subject
to the weaknesses inherent in any hypothesis testing approach. Hypothesis
testing suffers from three main problems in the forensic arena. The first
problem is that hypothesis tests fail to incorporate relevant evidence, such
as the relative frequency of the recovered glass and the mere presence of
glass fragments. The second problem is what Smalldon (personal communication, 1995) termed the “fall off the cliff” effect. It seems illogical that if
a probability of 0.989999 is returned from a hypothesis test then it should
be deemed as a match when a probability of 0.990001 would be a nonmatch,
particularly when these probabilities are calculated under distributional
assumptions, which (almost certainly) do not hold. The third problem is that
hypothesis testing does not answer the question of interest to the court.
Robertson and Vignaux106 argue that presentation of a probability answers
the predata question — “What is the probability of a match if I carry out
this procedure?” — rather than the postdata question — “How much does
this evidence increase the likelihood that the suspect is guilty?” It is, of
course, the latter that the court is interested in.
The suggestion to take a Bayesian approach to these problems is by no
means novel,5 and the continuous extension has been used in dealing with
RI-based data.8 The next section will extend the continuous approach for
multivariate (elemental composition) data.
3.6.1.1 The continuous likelihood ratio for elemental observations
Walsh et al.8 discuss a case where a pharmacy window was broken. Fragments of glass were retrieved from two suspects and compared with a sample
of fragments from the crime scene on the basis of mean RI. Both recovered
samples failed their respective t-tests and that is where the matter might
have ended. However, there were a number of aspects contradictory to the
conclusion that the fragments did not come from the crime scene window.
Both offenders had a large number of fragments of glass on their clothing.
Studies104,107,108 have shown that large groups of glass fragments on clothing
is a reasonably rare event on people unassociated with a crime. Examination
suggested that the recovered fragments came from a flat float glass object
— again a relatively rare event — and paint flakes recovered from one of
the suspects were unable to be distinguished from the paint in the window
frame at the crime scene.
Thus, the weight of the evidence supports the suggestion that both
suspects were at the crime scene when the window was broken. This is a
classic example of Lindley’s paradox,109 although Lindley himself refers to
©2000 CRC Press LLC
it as Jeffreys’ paradox. Although the samples failed the t-test, the results are
still more likely if they had come from the same source than from different
sources.
Walsh et al.8 propose an extension to the ideas put forward by Evett and
Buckleton 7 which retains grouping information while dropping the
match/nonmatch approach. The formula under consideration in their specific case is
LR " T0 +
where
TL P0 f ( X & Y | SX , SY )
ˆ
PS
1 L g(Y )
(3.44)
TL = the probability of three or more glass fragments
being transferred
P0 = the probability of a person’s having no glass on
their clothing
P1 = the probability of a person’s having one group of
glass on their clothing
SL = the probability that a group of glass on clothing
contains three or more fragments
–
ĝ (Y ) = the value of the probability density for float glass
at the mean of the recovered sample, usually
obtained from a density estimate
–
–
f(X – Y |S X, S Y) = the value of the probability density for the difference of two sample means; this is simply an
unscaled t-distribution using Welch’s modification
to Student’s t-test
The term T0 is generally small and, hence, can be dropped without
significant loss in generality or accuracy. Therefore, Equation 3.1 can be
rewritten as
LR " T0 +
TL P0
.lrcont
PS
1 L
(3.45)
where
lrcont "
(
f x & y |Sx , Sy
gˆ ( y )
)
(3.46)
In a case where the glass evidence is quantified by elemental decomposition rather than by RI, the only change in evaluating the LR is the method
for evaluating lrcont.
©2000 CRC Press LLC
Hotelling’s T2 is a multivariate analog of the t-test that examines the
standardized squared distance between two points in p-dimensional space.
These two points, of course, are given by the estimated mean concentration
of the discriminating elements in both samples. It seems logical that the
– –
multivariate form of lrcont should replace f(X – Y |SX, SY) with the unscaled
probability density function for the distribution of T2. This, however, is not
quite as simple as it sounds. If there are nc control fragments and nr recovered
fragments to be compared on the concentration of p different elements, and
nc + nr > p + 1, then it is assumed that (1) the elemental data of the whole
of the control population (window/container/bottle) and the whole of the
recovered population is multivariate normal and (2) both populations have
the same covariance structure,89 i.e., the individual variances of the elements
are the same in each population, and the correlation between any two elements is the same in each population. If these assumptions are true, then T2
has an F-distribution scaled by the sample sizes, i.e.,
T2 ~
(nc + nr & 2) p
(nc + nr & p & 1)
Fp,nc +nr & p&1 = c 2 Fp,nc +nr & p&1
(3.47)
where
c2 =
(nc + nr & 2)p
(nc + nr & p & 1)
(3.48)
–
–
Thus, f(X – Y |SX, SY) should be replaced by the value of the unscaled
probability density for an F-distribution on p and nc + nr – p – 1 degrees of
–
freedom at T2/c2, and ĝ(Y ) should be replaced by the value of a multivariate
–
–
probability density estimate at the recovered mean Y (recall that Y is now
a p × 1 vector). However, the scaling factor in the multivariate case is a matrix,
–
while both f(T2/c2) and ĝ(Y ) are scalars, so lrcont could be evaluated, but the
result would be a matrix and have no intuitive meaning. The solution to this
problem comes from the way the Hotelling’s T2 test works.
Hotelling’s T2 finds the linear combination of the variables that maximizes the squared standardized distance between the two mean vectors.
–
More specifically, there is some vector l (p × 1) of coefficients such that lT (X
–
2
– Y ) quantifies the maximum population difference. That is, if T rejects the
– –
null hypothesis of no difference, then lT(X – Y ) will have a nonzero mean.88
This fact provides the solution. If l can be found (and it can, see Appendix
A), then it can be shown that a new statistic tl2 (see Appendix A for the
definition) has the same distribution as T2,88 but has a scaling factor that is
– –
not a matrix. The numerator of lrcont, therefore, becomes f(lT(X – Y )|s l). The
numerator is the height of a probability density for an F-distribution on p
and nc + nr – p – 1 degrees of freedom at tl2/c2 transformed back to the scale
©2000 CRC Press LLC
of the linear combination. The denominator must be on the same scale, thus
–
it becomes the value of a univariate probability estimate density at lTY .
3.6.1.2 Examples
The data in the following examples come from two distinct sources, one
green bottle and one colorless bottle taken from the same plant at the same
time. Ten fragments were taken from each bottle, and the concentrations of
aluminum, calcium, barium, iron, and magnesium (p = 5) were determined
by ICP-AES. The quantities, TL, P0, P1, and SL, are taken to be those given in
Reference 8, so that
LR " 8lrcont
(3.49)
The first example uses five fragments from the green bottle as a control
sample (nc = 5) and five fragments from the same bottle as a recovered sample
(nr = 5) so that the population means are truly equal. lrcont " 2600, so in this
case the evidence would be 20,800 times more likely if the suspect was at
the crime scene than if he was not.
The second example takes the ten fragments from the green bottle as the
control sample (nc = 10) and the ten fragments from the colorless bottle as
the recovered sample (nr = 10), so the null hypothesis is false, i.e., the population means are truly different. lrcont " 4 × 10–10. Because the lrcont is so small,
the term T0 in Equation 3.45 now determines the LR. If T0 is taken at a typical
value of around 0.1, then in this case the evidence would be ten times less
likely if the suspect was at the crime scene than if he was not, i.e., the
evidence against the suspect strongly disputes the hypothesis that the suspect was at the crime scene.
3.6.1.3 Discussion
Hotelling’s T2 test for the difference in two mean vectors provides a valid
statistical method for the discrimination between two samples of glass based
on elemental data. However, it is subject to small problems and does not
answer the real question adequately. The Bayesian approach along with the
continuous extension is the only method that fulfills the requirements of the
forensic scientist, the statistician, and the court. All analyses of elemental
data should use the continuous Bayesian approach.
3.7 Summary
In this chapter we have presented the laws of probability and introduced
the Bayesian approach as the best available model for forensic inference. We
have presented a hierarchy of propositions, which could be addressed by
the court and/or expert. In order to illustrate the interaction of the various
parameters that are important in assessing glass evidence, we have described
©2000 CRC Press LLC
four hypothetical cases. For these cases a compromise approach has been
adopted, and because of the complexity of the continuous LR it is only at
the end of the chapter that we have presented a full Bayesian approach.
The framework presented has many advantages, one of which is to
highlight the sorts of surveys and experimental data that are needed to assess
the value of glass evidence. Most of the parameters appearing in the LR are
intuitively important to glass examiners, and some had been studied before
it was shown that Bayesian inference provided a logical and coherent framework for transfer evidence. However, the great majority of these studies,
particularly the surveys done on glass found on clothing at random, are
posterior. These surveys will be the subject of the following chapter.
3.8 Appendix A
We require
(
) = p(x & y , x , s , s |C)
p( x , y , s , s |C )
p( x , y , s , s |C )
p( x & y |x , s , s , C) p( x |s , s , C) p( s |s , C ) p( s |C)
=
×
×
×
p( y |x , s , s , C )
p( x |s , s , C ) p( s |s , C ) p( s |C )
p( x & y | x , s , s , C )
=
p( y |x , s , s , C )
p x , y , sx , sy |C
x
x
y
x
y
y
x
x
y
x
x
x
y
x
y
y
x
y
x
y
y
y
y
y
(3.50)
By assuming p(x– – y– |x– , sx , sy , C) ( p(x– – y– |sx , sy, C), we assume that the
sampling variation for fragments with mean µ is independent of µ:
(
) (
)
p y |x , sx , sy , C ( p y |C and
(
) (1
p( s | s , C )
p sx | s y , C
x
(3.51)
y
While this assumption is unlikely to be exactly true, the expected value
of this ratio is expected to be greater than one. Replacing it by one is unlikely
to disadvantage the defendant in the majority of cases.
We are left with the task of assessing
LR =
©2000 CRC Press LLC
(
p X & Y |X , Sx , Sy , C
(
p Y |X , Sx , Sy C
)
)
(3.52)
Options for the numerator include density functions such as Student’s
t-test and Welch’s modification of this.
To overcome the problem of false discrimination because of poor variance estimates, we advocate the use of Welch’s modification to the t-test.
This makes allowance for the variance of the recovered group and the control
group to be unequal.
©2000 CRC Press LLC
chapter four
Glass found at random and
frequency of glass
In Chapters 2 and 3, we looked at the two-stage, as well as the continuous,
approach to glass evidence interpretation. The two-stage approach is still
the most commonly used and proceeds by testing whether we should discriminate between the samples using various classical significance tests. If,
after this stage, we have failed to discriminate between the control and
recovered samples, it is necessary that we give a measure of the evidential
value of the match. This, we know intuitively, is inversely related to the
probability of a chance match given that the suspect is innocent; but as we
have seen see in Chapter 3 it is also influenced by other parameters such as
the prevalence of glass found at random and the probabilities of transfer
and persistence. To estimate these parameters we could, of course, content
ourselves with a personal opinion based on experience, but for several reasons it would be preferable to base our assessment on an appropriate body
of data. One of these reasons might be the recent ruling in the U.S. suggesting
that such expert opinion testimony may not be admissible unless the expert
can demonstrate that opinion is indeed scientific and not an art. It is necessary to be precise by what is meant by an “appropriate” body of data. Surely
it is a collection that enables us to address the questions that are relevant
given the circumstances of the offense and the suspect’s alleged involvement.
4.1 Relevant questions
It is much easier to talk in the abstract about “relevant questions” than it is
to identify them in a practical situation. The reason for this is simple — it
follows from one of the principles of many legal systems that the suspect is
under no obligation to offer an explanation for the presence of glass on his
clothing. If he does, then the questions may become easier to identify. For
example, if he says that he had recently been nearby when someone smashed
a milk bottle then we could seek a database of milk bottles. This, in itself, is
a simplification. The fact that the suspect has offered a possible explanation
©2000 CRC Press LLC
does not remove his right to the “best” defense. Interpreting this is not trivial,
but it might be that the alternative offered should be phrased as, “The glass
is from a milk bottle or another source that I cannot identify, unrelated to
the crime of which I am innocent.” Strictly, in such a case we require a survey
of the clothing of persons who have recently broken a milk bottle. Such a
specific survey will almost never be available. In general, however, no alternative will have been offered, and we face the task of setting questions which
must meet two, possibly conflicting, requirements: they must be relevant,
given our limited knowledge of the circumstances, and they must anticipate
our ability to answer them. The latter is determined by the availability of
data.
4.2 Availability
Creating a good data collection takes time and absorbs precious resources.
It has been tempting then, rather than to carry out custom-designed surveys,
to use a smaller amount of effort to abstract and organize data which come
our way as part of casework. Historically, the most obvious way to do this
has been through control data collections though, more recently, valuable
data have been collected from clothing encountered in casework.
In the next sections we give a brief historical review of forensic glass
data collections. We present the studies done on glass found at random
(clothing surveys) and on the distribution of analytical characteristics.
4.3 Glass found at random (clothing surveys)
The presence of glass found at random and the frequency of the analyzed
characteristics (generally RI) are two parameters that are related and difficult
to separate. They are often both studied in the clothing surveys presented
later, but we have chosen to treat them in different sections as we did in
Chapter 3. The phrase “clothing survey” will be used to cover any survey
which has looked for glass in connection with any part of the “person”: so
included here are surveys of footwear and hair combings. Two types of
populations have been studied: the general population and persons suspected of crime. A summary of the main clothing surveys is given in Table
4.1.
4.3.1 Glass found on the general population
4.3.1.1 Glass recovered on garments
In 1971, Pearson et al.110 published a survey of glass and paint fragments
from suits submitted to a dry cleaning establishment in Reading, England.
One hundred sets of jackets and trousers (described as suits) were examined,
and debris samples were collected from pockets and pant cuffs. Glass fragments were found in 63 of the suits examined. A total of 551 glass fragments
were recovered; 253 fragments, about half of the particles recovered, were
©2000 CRC Press LLC
Table 4.1
Summary of Clothing Glass Data Collections
Ref.
Date
Authors
110
1971 Pearson et al.
111
1977
112
1978
105
1985
107
1987
108
1992
104
1995
113
1994
114
1997
Country
Type of survey
Size
U.K. (FSS)
Clothing from a dry 100 suits
cleaning
establishment
Davis and
U.S.
Men's footwear
650 pairs
DeHaan
donated to charity
Harrison
U.K. (FSS)
Footwear from
99 shoes
casework
Harrison et al. U.K. (FSS)
Clothing from
200 cases
casework
McQuillan and U.K. (Belfast) Hair combings from 100 people
McCrossan
people unconnected
with crime
McQuillan and U.K. (Belfast) Clothing from
432 items of
Edgar
individuals
clothing
unconnected with
crime
Lambert et al. U.K. (FSS)
Clothing of people
589 people
suspected of breaking
offenses
Hoefler et al. Australia
Clothing from
47 sweatshirts
individuals
unconnected with
crime
Lau et al.
Canada
Clothing of people
48 jeans
unconnected with
213 students
crime
found on two suits, with the largest number being 166 fragments on a
singlesuit. Results were presented showing the distribution of the glass
among the suits and the size of the recovered fragments: about 3% were
larger than 1 mm, with the great majority (76%) measuring 0.1 to 0.5 mm.
These results are similar to those obtained by McQuillan and Edgar108 who
found that 5% of the fragments recovered in the pockets measured more
than 1 mm and 32% more than 0.5 mm (compared to 0 and 6% for particles
recovered on the surface).
There was no attempt in Pearson et al.110 to assess the number of groups
of glass present on the suits.
A significant rework of these glass samples, however, was undertaken
by Howden.55 His results showed that “the RI distribution on each suit
appeared to be random.” This will form an important point in our future
discussions. Therefore, it warrants both clarification (the sentence quoted is
not quite self-explanatory) and experimental support. Howden remeasured
the RIs of available “suits survey” glass fragments (approximately 400 of the
original 551 were still available). A group of glass was found to have an RI
near 1.5230, which was the same as glass Petri dishes in which the glass had
been stored. These fragments were omitted from further work. The remain-
©2000 CRC Press LLC
ing glass from each suit was described as random in that it did not form
groups of fragments with similar RIs. Howden comments that this is particularly obvious for those suits with large numbers of glass fragments on them.
Here, we formalize this as a postulate about glass on clothing of persons
unrelated to crime. Specifically, if glass is present it tends to be as small
groups, and large groups of glass are rare on such clothing. Both the word
large and rare need to be defined; however, this is possible by investigation
of the data that has been developed from survey work, and we will make
an attempt to do this in this chapter.
With hindsight, a flaw exists in Pearson et al.’s survey. The debris collected was from the pant cuffs, pant pockets, and jacket pockets. Therefore,
it appears that debris from the surfaces of the clothing was not collected.
This flaw may be of minor significance. Recently, evidence interpretation has
emphasized the significance of glass on the surfaces and particularly the
surfaces of the upper clothing. Since the suits survey does not examine these,
it cannot be used in this way. However, the very significance of the finding
of glass on the surfaces is because there is not a lot of glass on the surfaces
of clothing. Therefore, the suits survey may have sampled most of the glass
that was on the clothing.
In Hoefler et al.’s113 study on the presence of glass at random on the
general population in Australia, only the surface of the garments was examined. The research showed that 41% of the garments had no glass, 53.8%
had between one to eight fragments, and surprisingly 5.2% (that is five
garments) had more than ten fragments. No attempt was made to assess the
number of groups. The majority of the recovered particles were very small,
around 0.1 mm. These results corroborate the data provided by Pearson et
al.110 and McQuillan and Edgar.108
In 1997 Lau et al.114 investigated the presence of glass and paint in the
general Canadian population. The clothing of 213 students was examined:
1% of upper garments and 3% of lower garments bore fragments on their
surface. One fragment was found on five out of six garments, and on the
sixth garment two particles were recovered. All fragments were smaller than
1 mm2, and only one pocket yielded a glass particle. These results differ from
the other studies presented previously, as very few fragments were recovered. This may be because of the type of clothing studied (mainly T-shirts),
the population, and/or the searching methods.
4.3.1.2 Glass recovered on shoes
Davis and DeHaan111 examined embedded particulate debris from 650 pairs
of men’s footwear which had been donated to a charity in Sacramento, CA.
Only 20% of the shoes were found to contain colorless glass fragments. The
total number of recovered glass fragments was not recorded, but it was noted
that of the 261 individual shoes containing glass, 215 contained fewer than
four fragments and the remaining 46 contained four or more fragments.
Inspection of the histogram for the size distribution of the fragments suggests
that a total of about 450 colorless glass fragments were recovered. The Lau
©2000 CRC Press LLC
et al.114 survey does not corroborate this study; indeed, on the 164 pairs
examined only eight fragments were found on the soles (5% of footwear
presented glass). No glass was embedded and no glass was found on the
upper part of the shoes. As most particles recovered in Davis and DeHaan111
were smaller than 0.84 mm and small fragments are difficult to recover
without a probe, searching methods may again explain the difference
between the two studies. The populations studied were also different.
4.3.1.3 Glass recovered in hair
McQuillan and McCrossan107 looked for the presence of glass in hair combings (Table 4.2). In samples from 97 friends and relatives of the staff at the
Northern Ireland Forensic Science Laboratory, only one glass fragment,
smaller than 1 mm, was recovered. No glass was recovered from the hair
combings of 20 motor mechanics, and a total of only eight fragments was
recovered from a group of six glaziers. No glass was found in the hair
combings of two of these six glaziers, one glazier had five fragments coming
from four different sources, and the remaining three glaziers had a single
fragment (see Table 4.3). These results highlight the evidential value of glass
recovered in hair combings.
Table 4.2 Glass Recovered in Hair in Three
Different Populations107
Type of population
% of persons having
glass in their hair
“Normal”
Mechanics
Glaziers
1/97
0/20
4/6
Table 4.3 Glass Recovered in the Hair of
Glaziers107
Glazier
number
Number of
fragments
Number of
sources
1
2
3
4
5
6
0
0
5
1
1
1
0
0
4
1
1
1
4.3.2 Glass recovered on the suspect population
4.3.2.1 General trends
In 1985 Harrison et al.105 reported a survey of glass on clothing of persons
suspected of involvement in crime. The survey consisted of about 200 cases,
and normal U.K. practice was followed with respect to recovery of the glass
and selection of fragments for examination. The RI of more than 2000 glass
©2000 CRC Press LLC
fragments was measured; 40% of these fragments were found to match the
respective control samples. Glass was classified as having originated from
hair, surface of clothing, pant pockets and cuffs, upper parts of shoes, and
embedded in soles. As shown in Table 4.4, the highest proportion of matching
glass was found in hair and the lowest proportion was found embedded in
shoes.
Table 4.4 Distribution of Recovered Fragments from Harrison et
al.105 and Lambert et al.104
Location
Hair
Clothing/surface
Pockets/cuffs
Upper parts of shoes
Embedded in soles
Harrison et al.105
Lambert et al.104
% of total matching % of total matching
69
54
32
47
9
75
54
36
43
N/A
We draw some inferences from this. It is incumbent upon forensic scientists to be fair to the defendant. This might, we feel rightly, be construed
to mean that they should not expose him/her to unnecessary risk of random
matching. Since most “random” glass is found in the soles of the shoes, this
suggests a “top down/outside in” search strategy with the search being
terminated when a “large” (yet to be defined) amount of glass has been
found. Such a strategy would put hair combings as the first to be searched
followed by the surfaces of the upper clothing; next come the surfaces of the
pants, shoes, and pockets; and last would be the soles of the shoes. Doing
elemental analysis with SEM-EDX, it was found that the majority of the
nonmatching fragments large enough to be analyzed did not seem to come
from windows.54,55
4.3.2.2 Glass recovered on shoes
In a previous study, Harrison,112 following a serious case of criminal damage
in Newcastle-upon-Tyne, searched for the presence of glass on 99 shoes (49
pairs and one odd shoe) from ten suspects (therefore, a maximum of ten
pairs of shoes can be “guilty” and the remainder should be “innocent”).
Debris was collected from the sole, heel, upper parts, inside, and wrappings
of each shoe. No glass was contained in 15 of the 99 shoes. In Lambert et
al.104 the number of shoes having no glass was a little lower; on 402 shoes
studied, 31% (compared to 15%) had no glass, 12% had matching glass, 23%
had matching and nonmatching glass, and 34% had nonmatching glass only.
Therefore, it was not unusual to find glass on shoes, and it is worthwhile
noting that more than half of the recovered glass fragments did not match
the control.
The distribution of group sizes is most like the distribution on glass
recovered on pockets or surface: the size of the groups is generally 1 fragment
(in more than 80% of the cases).
©2000 CRC Press LLC
It is more common to recover glass on a suspect population than on the
general population. However, the general findings of both types of surveys
corroborate the hypothesis that glass recovered on soles has less evidential
value than glass recovered on the surface of clothing.
4.4 Comparison between suspect and general populations:
an example
We are now going to concentrate on two surveys. The first survey was carried
out on people unconnected with crime in Northern Ireland by McQuillan
and Edgar.108 We will refer to this as ME. The other survey, reported by
Lambert et al.,104 was a large casework study done as a follow-up to that
reported by Harrison et al.105 We will refer to this as LSH. We will explore
the extent to which the surveys provide information on the numbers of
groups of glass fragments found and the sizes of the groups of fragments.
The ME study was of 432 garments from individuals who had no suspected involvement in crime: members of a youth club, the Ulster Defense
Regiment, and the Royal Ulster Constabulary. For each person a pair of
garments was examined, i.e., pants with either a jacket or a sweater. They
found that 39% of the pant/jacket pairs (including pockets) bore no glass —
a proportion which was very close to that reported by Pearson et al.110 and
Hoefler et al.113 in their surveys.
The LSH survey was set up as follows. During a period of 1984/1985,
the six laboratories of the FSS and the Northern Ireland Forensic Science
Laboratory collected data on both matching and nonmatching glass that they
encountered in searching clothing as part of normal glass casework. Items
were collected from 589 individuals, and the RIs of over 4000 fragments were
measured. There is a good rationale behind this sort of approach. If, in each
of the surveyed cases, we attribute any glass that matches the case’s controls
as positive evidence for that particular investigation, then any remaining
nonmatching glass can credibly be viewed as background material unconnected with any particular incident. Furthermore, without making any contentious assumptions about the guilt or innocence of the casework subjects,
it is indisputable that they are people who have come to police notice in
connection with the investigation of breaking offenses. Zoro and Fereday115
have convincingly argued that this is the relevant population to be considered when considering the suspect as an innocent person.
In Figure 4.1 we look at the numbers of groups of glass found per
individual and compare the findings of the two surveys. Remember, (1) in
the LSH survey we consider only the nonmatching glass, and (2) we are
considering surface fragments found per individual, i.e., found on an
upper/lower pair of garments. The proportion of garment pairs with no
glass on their surface was higher in the ME survey (64% compared with
42%). Otherwise, the two distributions have similar shapes. In the LSH
survey the groups are smaller than in the ME survey (see Figure 4.2).
©2000 CRC Press LLC
70
Percentage of
individuals
60
50
40
LSH
30
McQ
20
10
0
0
1
2
3
4
5
6
7
8
Number of groups found
Percentage of groups
Figure 4.1 Number of groups of glass found per individual (surface only) in the
LSH and McQ surveys.
100
90
80
70
60
50
40
30
20
10
0
LSH
McQ
1
2
3
4
5
6
7
Number of fragments per group
Figure 4.2
surveys.
Distribution of the size of groups of fragments in the LSH and McQ
4.5 Estimation of the probability of finding at random i
groups of j fragments
Tables 4.5 and 4.6 give a summary of the data from the ME survey in a
form that can be used in the type of calculations we are advocating — that
we intend to act as a resource in the practical application of these statistical
©2000 CRC Press LLC
methods. It would be preferable if each country could perform its own
survey; however, we realize the difficulties and costs in this. The ME data
has been summarized for size of fragments into small groups (one to two
fragments) and large groups (three or more fragments), as this was the
thinking at the time. The choice of the definition of small and large is
arbitrary and subjective. We hope that a study of the survey data by other
investigators might lead to similar conclusions. At that time our standard
practice was to ask people to subjectively estimate the probability of transfer
of a large or small group of glass. These coarse divisions were about all that
could be handled for transfer estimation at that time, and it was necessary
to align the S terms in a similar way. It was not until the development of
the graphical models (vis-à-vis TFER, a program to assemble estimate transfer probabilities from data, discussed in Chapter 5) that it was realistic to
try to model the transfer of 0, 1, 2,…, etc. glass fragments.
Table 4.5 The Number of Groups of Glass Found for Different Search Strategies from
the ME Survey
Number of Upper garments Upper and lower Upper garments Upper and lower
groups of glass surface only
garments
surface and
garments
surface only
pockets
surface and
pockets
P0
P1
P2
P3
P4
P5
P6
P7
P8
P9
0.811
0.146
0.029
0.000
0.010
0.005
0.000
0.000
0.000
0.000
0.636
0.238
0.087
0.010
0.010
0.005
0.000
0.005
0.000
0.000
0.641
0.180
0.053
0.063
0.024
0.015
0.000
0.000
0.015
0.005
0.403
0.272
0.087
0.053
0.092
0.015
0.019
0.005
0.019
0.015
Note: Data have been grouped using the Evett and Lambert grouping algorithm. Data were
reworked by Buckleton and Pinchin from the raw data and differ slightly from the
published set.
Table 4.6 The Size of Groups of Glass Found for Different Search Strategies from
the ME Survey
Size of groups of Upper garments Upper and lower Upper garments Upper and lower
glass
surface only
garments
surface and
garments
surface only
pockets
surface and
pockets
1 or 2 fragments
3 or more
0.980
0.020
0.971
0.029
0.958
0.042
0.965
0.035
Note: Data have been grouped using the Evett and Lambert grouping algorithm. Data were
reworked by Buckleton and Pinchin from the raw data and differ slightly from the
published set.
©2000 CRC Press LLC
The data in the large and small format are not as directly useful when
assessing, say, S6 as they could be, but they are useful adjuncts to subjective
judgment.
4.6 Frequency of the analyzed characteristics
At the beginning of this chapter we mentioned that most studies on glass
found at random have also considered the frequency of physical and/or
chemical characteristics of the recovered glass. This is very important
because the frequency of the analyzed characteristics is of interest only if the
glass is present by chance and, therefore, does not come from the control.
This conclusion follows logically from the Bayesian approach, but only more
recently has been accepted. The first databases used to estimate the frequency
of the analytical characteristics included mainly glass coming from controls.
Tables 4.7 and 4.8 show a summary of the studies that will be presented.
Table 4.7
Ref.
25
116
16
19
117
Summary of Control Glass Data Collections
Date
Authors
Country
1968 Cobb
U.K. (FSS)
1972 Dabbs and
Pearson
1977 Lambert and
1984 Evett
1982 Miller
U.K. (FSS)
Type of survey
Size
Control glass from 175 controls
breaking cases in
the West Midlands
Fire survey data
939 samples
U.K. (FSS)
Control glass from 9000+ controls
casework
U.S. (FBI)
Flat glass from
1200 samples
casework
1986 Buckleton et al. New Zealand Glass samples from 513 samples
388 vehicles
1989 Faulkner and
U.K. (MPFSL) Samples of
300 samples
Underhill
nonwindow glass
Table 4.8 Summary of Clothing Glass Data Collections Where Analytical Data Were
Collected
Ref.
Date
Authors
110
1971 Pearson et al.
111
1977
112
105
108
1978
1985
1992
104
1995
Country
Clothing from a dry
cleaning establishment
Davis and
U.S.
Men’s footwear donated
DeHaan
to charity
Harrison
U.K. (FSS)
Footwear from casework
Harrison et al. U.K. (FSS)
Clothing from casework
McQuillan and U.K. (Belfast) Clothing from individuals
Edgar
unconnected with crime
Lambert et al. U.K. (FSS)
Clothing of people
suspected of breaking
offenses
©2000 CRC Press LLC
U.K. (FSS)
Type of survey
Size
100 suits
650 pairs
99 shoes
200 cases
432 items of
clothing
589 people
4.7 Control glass data collections
Initiatives to collect control glass examined in the course of routine casework
started in the 1960s and are now widespread. The information on the RI
distribution of control glass (information on elemental analysis is scarce)
shows that the frequency of the RI depends on the type of glass, its age, and
its geographical origin.19,86,91
Imagine that the examiner finds a group of fragments on the suspect’s
clothing that “match” the control from the scene. Let us assume that the only
collection available to us is a control data collection. In that case we are
driven to ask the following kinds of questions.
What is the probability that a group of fragments from some other glass previously submitted would have matched the recovered fragments?
This is the “coincidence probability” approach that we talked about in
Chapter 2. But, of course, we have problems straight away: what do we
mean by “some other glass source”? If we can break our data collection into
classifications such as “window,” “container,” etc., then which of these
should we use? As we have seen in Chapter 3 the frequency is only of interest
if the suspect did not break the window (or any other control object), and
in that case there is no justification for assuming that the recovered glass
must have come from another window. Resolution of this apparent conundrum was never achieved by any kind of theoretical consideration. Instead,
it became one of the main drivers of the move in the late 1970s/early 1980s
to routine use of elemental analysis of glass, largely because of its potential
for classifying glass fragments, and, similarly, it was one of the motivations
for the introduction of the interferometer.32,33
Let us assume that elemental analysis shows that the recovered fragments are, indeed, window glass. Then we can consult our control window
glass data and make the following statement.
There is a certain probability that the recovered fragments would match a
different window control.
If we assume that the control glass data collection is representative of
the distribution among broken windows in general, then this statement
undoubtedly has some value. The smaller the probability of such a coincidental match is, the stronger the evidence.
If there is no data suggesting that the recovered glass may have come
from a window pane, it would be erroneous to use control databases. Indeed,
clothing surveys, as well as miscellaneous studies, have shown that glass
broken at random originates mainly from sources other that windows. For
example, Zoro and Fereday115 reported the analysis of responses from a
random sample of 5000 members of the general public on their recent contact
©2000 CRC Press LLC
with breaking glass. Some general conclusions are that about a third of
respondents claimed to have broken or been close to breaking glass within
the previous seven days. Of these, most had only been in contact with one
source of broken glass, and most of the broken glass objects were not windows.
The same year Walsh and Buckleton118 reported a survey of glass fragments found on footpaths in different areas of Auckland, New Zealand. The
main aim of this survey was to estimate the proportions of the different types
of glass encountered. From 52 km of footpath, 1068 pieces of colorless glass
were collected. Footpaths were classified as being in central city areas, residential areas, light industrial areas, and areas where buildings were absent.
Container glass dominated in all areas, totaling about 70% of the glass
collected. Building windows represented about 12%, and most of the remaining glass was from vehicles. Even in the city areas only about 19% of the
glass was from buildings.
Control data collections only allow part of the job to be done. It constrains us to addressing a question that may not be the most relevant one
as far as the deliberations of a court are concerned. This has been recognized
for as long as data collections have been in existence. It seems reasonable to
take this body of data as a suggestion that control glass surveys are not
correctly balanced with regard to the type of glass, and that clothing surveys
should be used whenever possible.
4.8 Clothing surveys
In most clothing surveys presented earlier the physical characteristics of the
recovered fragments have been analyzed in order to see if the RI distributions
of glass recovered at random and control glass were different. Only occasionally has elemental analysis been used in order to clarify any putative
classification of the recovered particles.
4.9 Characteristics of glass found on the general population
4.9.1 Glass recovered on garments
Pearson et al.110 measured the RI of 551 fragments of glass recovered on suits
(see Table 4.8). The RI distribution was clearly different from that of the Cobb
data (see Table 4.7). Thus, the tentative conclusion expressed by the authors
was that about 30% of the glass was probably window glass. Howden55
studied this hypothesis doing elemental analysis on the fragments recovered
in Pearson et al.110 It was shown that most glass on clothing is not window
glass, but is composed significantly of tableware. There are important consequences if this hypothesis is true, and, therefore, it is worthwhile offering
some experimental support for it. Howden reexamined suitable and overlapping subsets of the fragments using various elemental techniques. He
found that
©2000 CRC Press LLC
Consideration of the refractive indices, atomic absorption, X-ray fluorescence and microprobe analysis results all lead to a similar overall conclusion. Namely
that much of the glass on clothing originates from tableware or colored glasses and that not much comes
from windows or colorless bottles.
If this latter hypothesis is true, then the presence of matching window
glass in the population studied, that is glass recovered on clothing that can
be shown to be window glass that matches a window control, may be very
evidential because window glass on clothing is rare. Such an interpretation
would be valuable and is worthwhile investigating and, if possible, quantifying. McQuillan and Edgar’s study shows comparable results.
4.9.2 Glass recovered on shoes
Davis and DeHaan111 measured the RI of 366 of the larger glass fragments
recovered in their survey. The RI distribution showed that a considerable
number of fragments had an RI greater than 1.533. The authors concluded
that most of the glass came either from containers or window panes. In the
study mentioned previously, Harrison112 analyzed the RI of 363 glass fragments recovered from 62 of the 77 shoes. The RI distribution was compared
with and found to be markedly different from the available control glass
data. Comparisons between the RI distribution of the glass recovered on the
shoes and the subsets of glass in the Home Office Central Research Establishment control data files showed that the best fit is to “container” glass.
This comparison also suggests that the shoes had more glass with low RI
(less than 1.516). This deficiency could be made up by increasing the contribution of “Container glass [sic] which is not bottle.” It seems plausible to
conclude from this comparison that the glass on the shoes was largely not
of window origin, but was most likely composed of container glass with a
significant contribution of tableware. This hypothesis is partly supported by
Walsh and Buckleton’s118 results.
Comparisons between Davis and DeHaan’s111 study in the U.S. and
Harrison et al.’s study105,112 in the U.K. show the same general tendencies.
The difference in the RI distribution, that has also been shown for control
glass, is explainable by the fact that the glasses found in the two countries
are likely to have come from different manufacturers and plants. The RI
distribution also depends on the date of the studies and is the plausible
explanation for the differences between Pearson et al.110 and Harrison et al.105
If we accept Harrison’s point that “clothing” surveys should be used
and we observe that control collections are dominated by window glass and
further we accept that glass on clothing is largely container glass dominated
by tableware, then again we see that control glass collections are of limited
use, except if a method has been used to classify glass. Even then, they will
answer only part of the question.
©2000 CRC Press LLC
4.10 Characteristics of glass found on the suspect population
In their study on glass found at random on persons suspected of crime,
Harrison et al.105 produced RI frequency distributions for the raw and
grouped nonmatching glass data. These were compared with the published
control glass data and shown to be generally different. It appeared that a
considerable proportion of the nonmatching glass originated from sources
other than windows. For example, the RI distribution of glass embedded in
the soles of footwear agreed with the RI distribution of control container
glass. A very limited amount of elemental analysis on the nonmatching
recovered glass suggested that about a third of the glass in the RI range
typical of windows had originated from a nonwindow source. The RI distribution of the matching glass in the survey was, not surprisingly, similar
to that of the control glass data.
It is tempting to argue that even some of the “nonmatching glass” is
actually matching and, therefore, of largely window origin. This allegation
could be made because of the propensity of grouping criteria to “drop-off”
outliers. If this were true then it would suggest that the estimate of Harrison
et al. of the fraction of window glass in their nonmatching survey is an
overestimate. However, J. Lambert (personal communication, 1995) has convincingly replied to this argument by showing that the distribution of nonmatching glass on people with only nonmatching glass is not different from
the distribution of nonmatching glass on people who also have matching
glass. In view of this, it seems reasonable to accept Harrison et al.’s conclusion.
4.11 Comparison between suspect and general populations:
an example
Figure 4.3 shows the distribution of RI measurements for the groups of
nonmatching fragments in the LSH survey. The ME distribution is not shown
in detail, but in Figure 4.4 we compare the general shape of the two distributions. There is reasonable agreement between the two. The higher proportion of LSH data in the range of 1.5150 to 1.5180 suggests a greater proportion
of window glass than in the ME survey. Since window glass is the most
common type to be broken during criminal activity, and since police tend to
arrest suspects who they believe engage in regular criminal activity, it is to
be expected that clothing obtained from such suspects would contain more
window glass than the clothing of members of the public who have no
involvement in crime.
However, the imagined need to discriminate between different types of
glass is not so immediate when we have a clothing survey to refer to.
©2000 CRC Press LLC
3
% FREQUENCY
2.5
2
1.5
1
0.5
0
5150
5160
5170
5180
5190
5200
5210
5220
5230
5240
5250
RI
Figure 4.3 RI distribution for groups of nonmatching glass in the LSH survey. ( The
RIs have been transformed by subtracting one and multiplying by 10,000.)
16
% FREQUENCY
14
12
10
8
6
4
2
0
5100
5120
5140
5160
5180
5200
5220
5240
5260
5280
Figure 4.4 Comparison of the RI distribution for nonmatching glass in the LSH
(———) and ME (— — —) surveys.
4.12 Summary
In conclusion, we would like to make a “philosophical” point. When asked
to fit a simple geometric shape to the “glass recovered from shoes” distribution or many other “glass from clothing” distributions, most glass analysts
choose a rectangle. The logical consequence of this “thought experiment” is
that most glass analysts feel that a reasonable approximation is to treat
frequency as a constant. This is of interest because so much effort has been
©2000 CRC Press LLC
put into assessing frequency, and it suggests that we could usefully turn our
attention elsewhere. Where? We would suggest that assessing the amount
of glass on persons and the probabilities of transfer are useful places for this
investment.
©2000 CRC Press LLC
chapter five
Transfer and persistence
studies
As we have seen in Chapter 3, background information such as transfer and
persistence is essential to evaluate glass evidence, and its importance cannot
be overstated. It is very important that the glass examiner be aware of the
data that have been published on the subject. If a search is performed, this
data is necessary to form an opinion on the value of evidence. Transfer and
persistence probabilities are very difficult for the expert to assess as they
may depend on various factors. The amount of glass expected to be recovered
is reported to be influenced by the type and thickness of the glass, the
distance at which the window has been broken, the garments of the suspect,
the time elapsed between the incident and the arrest, and the activities of
the suspect between the offense and the arrest. It may also be affected by
other variables such as the size of the window, whether or not entry was
gained to the premises, or the weather at the time of the incident.
Such information can assist in decision making. For example, given the
information supplied on the case submission form, would I expect to find
glass on the clothing of this suspect, and if so, where and how much? The
answers to these questions should guide the forensic scientist as to which
item should be examined first and which items should not be examined at
all. A survey of data from submission forms actually suggests that the data
in them may be of little use, which is a disappointing result for those advocating a rational approach to glass casework.119
The data on transfer is extensive. However, it is quite difficult to organize
due to a very large number of variables. This also makes it a difficult task
for the examiner to estimate transfer probabilities in any specific case. In
order to facilitate this, we introduce TFER, a program to assemble an estimate
from the data.
5.1 Transfer of glass
We begin by observing that most glass cases involve flexible glass (relatively
thin panes) that has been broken in flexion. It has been previously noted that
©2000 CRC Press LLC
the nature of the impacting object is not crucial to breakage under flexion.
It must have sufficient kinetic energy, but beyond that the exact nature of
the impacting object is thought to be, at least, secondary. Breakage is thought
to progress by the extension of radial cracks and by the propagation of
concentric cracks. Later in this chapter we will discuss the preponderance
of surface fragments in glass backscattered from breakage. Plausible hypotheses are available for this. An examination of glass pieces after breakage will
reveal many fragments missing from the edges of the cracks. It is reasonable
to believe that these fragments have been ejected during propagation of the
cracks by tension or compression forces.
Kirk120 first described the transfer of fragments onto the garments of a
person breaking a window, but it was not until 1967 that Nelson and Revell2
demonstrated the phenomenon scientifically by publishing photographs of
glass particles projected backward. Since then the mechanisms of transfer
and persistence have received some attention. The studies described in the
following sections provide, in general terms, three types of information to
assist in assessing the effects of the previous factors.
1. Glass smashing experiments, which examine the number and distribution of glass fragments transferred to the ground, involve smashing
different types of glass with different methods of breakage.
2. Glass smashing experiments that look at quantity of glass fragments
transferred to individuals who are nearby. Work on the influence of
parameters such as the weather or the breaking device, as well as
research on secondary transfer, is included in this group.
3. Studies on the retention of glass fragments on clothing with respect
to time and subsequent activity.
5.1.1 Transfer of glass to the ground
5.1.1.1 Number, size, and distribution of the fragments
Because of the complexity of the transfer process, it is necessary to control
as many parameters as possible. Nelson and Revell,2 Locke and Unikowski,121,122 and Locke and Scranage30 proposed standardizing the breakage
of windows with a pendulum.
Nelson and Revell’s study2 was performed on 19 panes of 0.6 × 0.9 m
glass. Two thicknesses (3 and 6 mm) were used, and the panes were broken
either with a pendulum or a hammer. The results of their research are as
follows.
1. Fragments were projected as far as 3.3 m.
2. A stroke in the middle of a pane produced more fragments than a
stroke in a corner.
3. The faster the pendulum, the smaller the hole and the fewer fragments
were produced.
©2000 CRC Press LLC
4. If the speed of the pendulum was high, then the dispersion (distance,
angle) of the fragments was reduced.
5. Although most of the backscattered fragments from 3 mm window
glass were flakes and chips, the fragmentation from 6 mm glass also
included an appreciable number of needle-like slivers.
6. Transfer is a variable phenomenon, especially when the windows are
broken with a hammer.
Their final conclusion was
In any investigation where window glass has been
broken it is well worthwhile — in fact almost a matter
of duty — for the investigator to secure samples of the
broken glass and the clothing of the suspect and submit them for laboratory examination.
This research revealed a number of patterns. However, Nelson and Revell performed only a small number of experiments and did not give quantitative data regarding the number and the size of the fragments transferred.
Locke and Unikowski121,122 and Locke and Scranage30 have carried out quantitative studies on transfer. The aim of the first study was to establish the
reproducibility of transfer and give an idea as to how many fragments are
transferred depending on the distance and the position of the person standing nearby. The apparatus used to break windows was standardized, and
the debris was collected in trays according to distance and direction. The
eight experiments demonstrated that transfer is a variable phenomenon even
when the conditions are standardized to a maximum. Nevertheless, it was
possible to show the following.
1. The number of fragments varied by a factor of 4.
2. This number depends on the distance and declined very rapidly.
3. Fragments smaller than 1 mm were homogeneously distributed in the
different sectors, but bigger fragments were mostly found in the inner
sectors.
Pounds and Smalldon,123 who employed less standard breaking procedure (the windows were broken with a hammer or a brick), obtained similar
results to Nelson and Revell (point 1) and Locke and Unikowski (points 2
and 3). Luce et al.124 also confirmed point 2 (Locke and Unikowski): the
number of fragments observed in their research declined by a factor of 4 to
5 for every 45 cm.
5.1.1.2 Influence of the window type and size
Having demonstrated that patterns could be revealed even in the face of
varying experimental conditions, Locke and Unikowski122 studied the effect
©2000 CRC Press LLC
of size and thickness of windows, as well as the influence of the type of
glass, on the number of fragments transferred. The characteristics of the 24
windowpanes are shown in Table 5.1.
Table 5.1
Types of Window Tested in Locke and Unikowski122
Size
Thickness
Wired
1 m × 1 m × 4 mm
1 m × 1 m × 4 mm 1 m × 1 m × 6 mm
0.5 m × 0.5 m × 4 mm
1 m × 1 m × 6 mm 0.5 m × 0.5 m × 6 mm
0.25 m × 0.25 m × 4 mm 1 m × 1 m × 10 mm 0.25 m × 0.25 m × 6 mm
Patterned
1 m × 1 m × 4 mm
0.5 m × 0.5 m × 4 mm
0.25 m × 0.25 m × 4 mm
The experiments, duplicated for each type of glass, showed that the
number of fragments does not depend on the size of the windowpane, but
rather on the total amount of cracking and the degree of disintegration.
Whether the window was patterned or not did not influence the number of
projected particles, but if the windowpane was wired or very thick, more
fragments were produced. This phenomenon could be explained by the fact
that more energy was needed to break wired or very thick glass, and, therefore, more energy was dissipated. This hypothesis was corroborated by the
observations of Luce et al.124 who suggested that force and number of particles are related. However, Nelson and Revell’s2 conclusions on the correlation between the number of fragments and the speed of the pendulum do
not necessarily support this hypothesis. In the case of wired glass, another
explanation would be that the particles are not only induced by radial and
concentric fractures, but also by fractures between the wire and the glass.
5.1.1.3 Presence of an original surface
As the presence of an original surface on a recovered glass fragment can
establish its origin (container or flat glass) and also influences its RI, it is
helpful to determine if fragments projected backward originate from the
interior, the exterior, or the bulk of the window. Following these observations, Locke and Scranage30 painted windows with black and red ink and
demonstrated that about 50% of the fragments measuring 1 to 0.25 mm came
from the surfaces. The exterior to interior surface ratio can vary from 3:1 to
30:1 depending on the experiment. The ratio also depended on distance: the
farther the person was from the window, the greater the fraction of fragments
that were projected from the rear. This could have been due to the fact that
secondary breaking is the most likely origin of fragments projected at a
distance. Since these fragments have the same chance of coming from the
interior or the exterior, the ratio is reduced (it tends toward 1:1). For particles
projected far from the window, a small number of particles is sufficient to
change the ratio. Locke and Scranage,30 Zoro,125 and Luce et al.124 obtain
comparable results. In experiments on glass transferred onto garments, this
tendency was also confirmed: indeed, Allen and others126-129 have shown that
60% of the fragments had an original surface, most of them coming from the
front of the pane.
©2000 CRC Press LLC
5.1.2 Transfer of glass broken with a firearm
Francis130 studied the backward fragmentation of glass as a result of an
impact from a firearm projectile. Three types of glass (ordinary clear house
glass, 3 mm thick; frosted wired glass, 5 mm thick; and clear laminated glass,
6.4 mm thick) and eight firearms and projectiles were tested. The panes were
broken at a distance of 4.0 m, with the shooter being directly in front of the
window. The fragments present on a grid measuring 4 × 4 m were then
counted in a manner similar to that employed by Locke and Unikowski121,122
or Luce et al.124 A large number of fragments were recovered; therefore, the
author recommended that “should a shooting occur through a window, the
suspect’s clothing, shoes and hair should be examined for the presence of
glass window.” No result has been given for the number of fragments recovered on the shooter. However, current research (T. Hicks, personal communication, 1999) is being performed in this area.
5.1.3 Transfer of vehicle glass
Locke et al.131 and Allen et al.132 have studied the transfer of vehicle glass.
In the first study, the authors investigated the nature of the fragmentation
process and the total number of fragments generated (tens of thousands of
casework size particles). In the second study, the authors examined the
distribution and number of glass particles transferred when breaking toughened and laminated screens. When the window was crazed, or had fracture
lines without significant breakage, over 1400 fragments in the range of 0.25
to 1.0 mm were found in the car. Although there was considerable variation,
it was shown that more particles were produced by shattering than by
crazing. By examining the spatial distribution, it was found that very few
fragments were present at distances greater than 1.5 m and at the sides of
the car.
5.1.4 Transfer of glass to individuals standing nearby
The work presented previously takes into consideration the number of fragments when distance, position, and glass type are varied. However, it does
not give an estimate of the number of fragments that can be projected onto
a person. In addition, it is uncertain whether the observations made on the
particles transferred to the ground are also applicable to fragments transferred to garments or hair.
The transfer of several types of glass (window, windscreens, and broken
glass) onto several receptors (garments, hair, and shoes) is presented next.
5.1.5 Transfer of window glass to garments
In order to standardize as many parameters as possible, Allen and Scranage
(5 experiments),126 Pounds and Smalldon (7 experiments),123 and Luce et al.
(8 experiments)124 have studied the transfer of glass when breaking a window
©2000 CRC Press LLC
with a pendulum. In order to estimate the number of fragments that may
be transferred in actual breaking, experiments have also been carried out on
glass broken with a tool or a stone.
5.1.6 Transfer of glass with a pendulum
Allen and Scranage126 performed five experiments breaking a 1 m × 1 m × 4
mm window with a pendulum. They found that the number of fragments
transferred to garments* was about 28 fragments at a distance of 0.5 m, 5
fragments at 1 m, 1 fragment at 1.5 m, and 2 fragments at 3 m. Most of the
fragments were found on the sweater, but substantial numbers were also
found on the socks. However, with greater distance, more fragments were
recovered on socks than on the sweater. Very few particles were found on
the pants. One windowpane was broken by a pendulum with a glove
attached in order to see how many fragments may be transferred to the
glove: 20 glass fragments were found embedded. Of these fragments, 18
were larger than 0.5 mm and, perhaps contrary to expectation, came predominantly from the rear surface.
The results of Luce et al.124 (8 experiments) show that the number of
transferred fragments varied between 19 and 14 recovered from the pants,
shoes, sweatshirt, and hair of a person standing at 0.45 m and 9 and 3 for a
person standing at a distance of 0.90 m. The variability of the results is not
surprising, given the variability in the number of fragments transferred to
the ground. But even if the variability is high, this research shows that
fragments are transferred to persons standing nearby and allows a comparison of results with the studies of glass transferred to the ground.
5.1.7 Glass broken under conditions similar to casework
In casework modus operandi is often unknown. Subsequently, many practitioners think that if it is unknown, then it is not possible to estimate the
number of fragments likely to have been transferred. We have found that
when subjects are asked to stand in a natural position and strike a window,
the positions taken are remarkably similar. Therefore, we believe that when
breaking a window the breaker can stand in three positions:
• He/she can charge the windowpane with the shoulder or the entire
body; the distance would then be 0.0 m.
• The breaker can strike the window pane with a tool or the foot; the
distance would then be approximately 0.6 to 0.9 m.
• He/she may throw an object; the distance would then exceed 1.5 m.
Therefore, we argue that even if the exact distance between the offender
and the window is unknown, it is possible to make some reasonable estimate.
The experiments presented next simulate the last two breaking procedures.
* Acrylic hat, polyester and wool socks, woolen sweater, synthetic pants, and leather gloves.
©2000 CRC Press LLC
Luce et al. (4 experiments),124 Pounds and Smalldon (4 experiments),123
Hoefler et al. (20 experiments),113 Hicks et al. (22 experiments, 15 breakings
with a hammer and 7 with a stone),133 Allen et al. (15 experiments),132 and
Cox et al. (15 experiments; personal communication, 1996) have studied the
transfer of glass resulting from the breakage of a window with a tool or a
brick. The aim of these studies was to determine if the number of fragments
transferred would differ from those obtained in the experiments with a
pendulum. Their results are summarized in Table 5.2.
Because of the different experimental conditions, it is difficult to compare
these results. First, different types of clothing have been used. For example,
Hoefler et al. have used 20 different jeans and sweatshirts, and Hicks et al.
have used a cotton tracksuit (person standing at 0.6 m) and one coarse
pullover and a pair of jeans (person standing at 0.8 m) throughout all experiments. Allen et al. and Cox et al. have used three types of clothing (shellsuits,
tracksuits, and a sweater and jeans). The size of the panes was also varied.
For example, the window panes used were 60 cm × 60 cm × 3 mm in Hicks
et al., a set of four panes having the dimensions 25 cm × 36 cm × 2.2 mm in
Hoefler et al, panes of different dimensions in Allen et al., and panes measuring 48 cm × 59 cm × 4 mm in Cox et al. Nevertheless, a study of Table
5.2 shows that more fragments are transferred with the hammer than with
the pendulum; the number of particles decreases with distance; and, according to Cox et al., the number of fragments initially transferred does not
depend (for the clothing used) on the type of garment. One can also observe,
with the most recent studies which replicate the experiments 15 or 20 times,
the high variability in the number of fragments transferred. As shown in
Table 5.2 this number was on average higher in Hicks et al.133 This difference
may be due to the breaking procedures, the force, the size of the panes,
or/and the natural variation. According to Hicks et al.133 and Luce et al.,124
on average more fragments would be produced when breaking the windowpane with multiple blows. The recovery procedure is also important and
may explain some differences. In all studies the fragments were recovered
by shaking, as recommended by Pounds,134 then separated into size categories and counted.
The way the subjects disrobed may have influenced the results. Hoefler
et al.113 and Allen et al.126-129 have shown that when disrobing, some glass
was lost (the range of fragments lost during disrobing was one to eight
particles in Allen et al.126-129). However, if we include the results of Cox et al.
on transfer of glass on upper and lower garments and compare all the studies
done on transfer, which of necessity are performed under disparate conditions, the most likely explanation for the variance is the inherent variability
of the transfer process in Figure 5.1.
Hicks et al. showed that 3 to 10% of the particles transferred were larger
than 0.5 mm. Luce et al.124 confirmed this result.
As a practical consequence of the studies, it is important, in casework,
to consider not only submitting clothing for examination, but also the sheets
on which the suspect has disrobed.
©2000 CRC Press LLC
©2000 CRC Press LLC
m
m
m
m
m
15
1
15
1
7
7
1
Pounds and Smalldon
Brick
Hammer (>1 strike)
Hammer
Hammer (>1 strike)
Brick
Stone
Crowbar (1 strike)
Allen et al.
Hicks et al.
Pounds and Smalldon
Hicks et al.
Pounds and Smalldon
Hicks et al.
Hammer
Hammer (1 strike)
Hammer (>1 strike)
Jemmy bar
Crowbar (1 strike)
Breaking device
Pounds and Smalldon
Luce et al.
Luce et al.
Hoefler et al.
Cox et al.
Research
—
M
M
M
L
M
M
M
L
M
M
—
H
—
M
H
—
Retna
Retention.
Note: The retention characteristics are described as low (L), medium (M), and high (H).
a
1.95 m
0.60
0.75
0.80
1.35
1.50
1
2
3
20
5
5
5
Repeats
Summary of Transfer Experiments
0.50–0.80 m
0.50 m
0.45 m
Distance
Table 5.2
27
12
50
12
39
27
26
12
16
15
127
11
40
4
4
5
3
Top
Mean
—
M
M
M
L
M
M
M
L
M
M
M
M
M
15–72
0–16
1–14
Retn
—
7–13
12–72
3–24
22–80
14–37
12–46
6–28
5–30
10–23
44–241
Range
4
6
24
—
2
3
9
27
27
25
9
16
13
31
Pants
Mean
—
1–3
1–3
3–22
13–38
16–41
14–36
5–15
6–29
8–29
5–81
—
4–85
—
0–10
0–16
—
Range
60
80
10
0
12
0
14
0
16
0
18
0
20
0
22
0
24
0
26
0
28
0
30
0
32
0
34
0
M
or
e
0
20
40
Frequency
12
10
8
6
4
2
0
Number of fragments
Figure 5.1 Number of fragments transferred on lower and upper garments at T =
0 when breaking a windowpane with a tool. (Results of studies by Hoefler et al.,
Hicks et al., and Cox et al.)
5.1.8 Transfer of vehicle glass and absence of glass
Rankin et al. (personal communication, 1989) have studied the transfer of
vehicle glass onto garments (the authors were investigating whether the
absence of glass could be an indication that a person had not smashed a
window). As a relatively small number of glass fragments were found on
the leather jacket used when reconstructing the case, it was concluded that
it was possible that no glass would be found given the case circumstances.
This is confirmed by the research done on transfer of windowpane glass,
where it has been shown that sometimes very little glass is transferred.
Therefore, the absence of glass is not an absolute indication that the suspect
has not committed the crime.* However, even this negative evidence warrants careful interpretation.
5.1.9 Transfer of glass when a person enters through a window
Allen et al.127 have studied the transfer of glass when breaking and stepping
through a windowpane. Three types of garments (shellsuit, tracksuit, and a
sweater and jeans) were used. It was found that rubbing the subject’s sleeve
against the broken edge of the pane had little effect on the number of
fragments transferred.
5.1.10 Influence of the weather on transfer
Allen et al.127 also investigated the influence of wetness of the clothing on
transfer using the following procedure: “The window was broken and the
dry clothes taken immediately. Fresh clothes were wetted with a hand
* This is a very general hypothesis. However, as seen in Chapter 3, which is devoted to the
Bayesian approach, there may be other hypotheses.
©2000 CRC Press LLC
sprayer, until the clothes were damp to touch. Another window was broken
(with a crowbar at a distance of 0.5 m) and clothing removed.” The conclusion of their study was that “Wet clothing (in particular shellsuits) may retain
larger numbers of fragments and a higher proportion of fragments greater
than 0.5 mm as compared to dry clothing.” However, as the number of
fragments is highly variable, this result must be taken with caution and
further research is necessary to confirm this result.
5.1.11 Transfer of broken glass
As mentioned earlier, Underhill135 compared glass transferred to clothing
during the breaking of a window with that transferred during clearing up
previously broken glass. More fragments (typically more than 100) were
transferred during clearing up, but a smaller proportion of these were found
to have original surfaces. Therefore, a high number of outer surface fragments would suggest backscatter. The relationship between fragment shape
(needle, chunk, and flake) and the mode of acquisition was investigated;
however, no significant correlation was found.
5.1.12 Transfer of window glass to hair
Several authors have studied the transfer of glass to hair or headwear. Glass
was either transferred by breaking a window with a pendulum, a tool, or a
brick or when clearing up the glass. As can be seen in Table 5.3, glass was
not always transferred, and, when there was transfer, the number was variable. However, again, these findings give an order of magnitude to the
number of glass fragments that one would expect to recover from hair a few
minutes after the breaking.
Table 5.3
Number of Fragments Recovered in Hair or Headwear
Approximate
distance
50 cm
80 cm
100 cm
140 cm
200 cm
Research
Pounds and
Smalldon
Luce et al.
Luce et al.
Luce et al.
Pounds and
Smalldon
Luce et al.
Allen et al.
Pounds and
Smalldon
Pounds and
Smalldon
©2000 CRC Press LLC
Breaking device
Number of Headwear or hair
experiments Mean # Range
Hammer
1
3
—
Hammer (1 strike)
Pendulum
Hammer (>1 strike)
Hammer
4
2
4
1
2
1
7
3
0–6
1
1–16
—
2
10
1
1
2
0
1
0–7
—
1
0
—
Pendulum
Crowbar
Brick
Brick
Underhill135 also reported the average number of fragments that were
recovered in hair after 10 minutes. Several experiments performed were
described as training exercises, backward fragmentation, and clearing-up
experiments. The windowpanes were broken at a distance of 0.60 m with a
crowbar. On average, the number of glass particles recovered was two in
clearing-up experiments and one when breaking the window. At most, 20
fragments were recovered in the hair (training exercises).
5.1.13 Transfer of window glass to footwear
No study has been published on glass transfer by kicking of the pane, which
is an obvious omission given the frequency with which items are submitted
from this modus operandi. Luce et al.124 have reported the number of glass
fragments that were transferred to footwear when breaking a windowpane.
On running shoes, 8, 0, 8, and 13 fragments were recovered when breaking
a windowpane with a single strike at a distance of about 0.50 m. In the four
experiments involving multiple hammer strikes, 14, 10, 22, and 47 particles
were recovered. An interesting finding was that more glass was found on
footwear than on pants or upper clothing when the person was standing 1.0
m from the breaking window. This result has been confirmed by Allen and
Scranage.126
5.1.14 Secondary and tertiary transfer
Bone136 describes secondary transfer experiments between two individuals
wearing woolen sweaters with very good retentive properties. Up to 500
glass fragments were placed on the front of the donor, who then hugged the
recipient for 5 to 60 seconds. Donor and recipient garments were then
removed and searched. In 12 experiments, the number of glass fragments
remaining on the recipient sweater after contact ranged from 0 to 9. A high,
though variable, proportion of glass was lost from the donor garment when
it was removed. In general, more fragments dropped off the recipient sweater
when it was removed than were retained on it. The contact time did not
have much effect on the percentage of glass remaining on the donor sweater,
but it did affect the number found on the recipient.
Bone’s research did not actually transfer glass by breaking a window.
Holcroft and Shearer137 conducted experiments where glass was transferred
to individuals by breaking a 4-mm thick pane with a hammer. In three repeat
experiments, an average of about 50 fragments were transferred to a retentive
sweater, and 20 fragments were transferred to jeans described as “retentive.”
Further experiments involved breaking the window followed by firm contact
with a second individual. In seven experiments 18 to 37 fragments remained
on the sweater and 9 to 19 fragments remained on the jeans. The recipient
individual wore a retentive sweater and fairly retentive cord pants. These
items were worn for either 5, 30, or 60 minutes before being checked for
©2000 CRC Press LLC
glass. For the seven experiments, two to six fragments were recovered from
the sweater, no glass was recovered (five times), and one and three fragments
were recovered from the pants. Two experiments with the donor and recipient wearing tracksuit tops with low retention properties resulted in four
and two fragments remaining on the donor item and no glass on the recipient
item.
Allen et al.128 also investigated secondary transfer. In their experiments,
they studied the possible transfer between a person who had broken a
window and a second person riding in the same car. Out of the 15 experiments involving the breakage of a windowpane with a crowbar, only one
fragment was found on the second person. Whereas, after the 20-minute
ride, typically 10 to 30 fragments were found (in one case 130 fragments
were found) on the breaker. No more than three particles in total were
transferred onto the car seat.
Allen et al.129 also studied the transfer of glass fragments from the surface
of an item to the person carrying it. Twelve experiments were performed
where a box was placed 1 m behind a window, which was subsequently
broken with a crowbar at a distance of 0.5 m. The box was then carried. In
all experiments glass was recovered on the box, and between 1 and 22
fragments were found on the upper/lower garments of the person carrying
it.
Tertiary transfer experiments were conducted by Holcroft and Shearer.137
The experiments involved placing 33 and 25 glass fragments on a chair,
followed by the recipient sitting for 5 minutes and walking for 5 minutes.
In each case no glass was found on the sweater and one fragment was
recovered from the jeans. A further experiment involved the breaker of the
window sitting in a chair for 5 minutes. Then the recipient also sat in the
chair for 5 minutes and walked around for a further 5 minutes before removal
and searching of the outer clothing. In three experiments the number of
fragments remaining on the sweater and jeans of the window breaker ranged
from 11 to 28 and 8 to 20, respectively. No glass was found on the sweater
of the recipient, and in each case one fragment was found on the jeans.
5.1.15 Transfer: what do we know?
The value of the published studies on the transfer of glass to garments is to
establish an order of magnitude for the number of fragments (0 to 100) that
can be transferred to a person breaking a window and to understand the
complexity of transfer. This depends not only on distance, but also on the
type of garment, the window, the force, or other unknown factors. They also
show that secondary transfer of a few particles is possible, but not probable.
These studies have shown that on average more fragments are recovered
on an upper garment that is at the height of the impact. Moreover, there
could be a relationship between the number of fragments recovered on the
©2000 CRC Press LLC
ground and on the garments:123 the number of particles smaller than 1 mm
recovered on a person would correspond with the number of fragments
found on the area of 300 cm2 where the person stood. This raises an interesting, but impractical idea that it would be possible to predict from the
crime scene the number of fragments transferred.
In summary, the conclusions on transfer are as follows.
1. It is possible to find glass on a person who has broken a window or
who was standing nearby.
2. Transfer is a variable process. The variability is hard to assess because
of the relatively small number of experiments performed and the
apparent high variability of the phenomenon itself.
3. The greatest distance at which particles can be transferred is about 3
or 4 m.
4. The order of magnitude of the number of fragments transferred is 0
to 100; usually the size of the fragments is between 0.1 and 1 mm.
5. The number of particles projected backward diminishes with distance:
every 0.20 m, the number decreases by half.
6. Experiments on the influence of the pane show that thick and/or
wired glass produces more fragments.
7. About 50% of particles transferred by backscatter have an original
surface.
8. Force (or pane resistance) may have an important influence on the
number of particles projected backward.
All the research published has been performed on the transfer of window
glass. However, transfer of other glass types are important, especially if the
suspect has an alternative explanation for the presence of glass on his clothing. Therefore, it would also be useful to study transfer of glass other than
from windows.
5.2 Persistence of glass on garments
Whereas early studies on persistence were scarce, there has been a considerable amount of research in several countries on the subject since the 1990s.
The two pioneering publications that are presented later used artificial transfer methods. This has led to some uncertainty regarding the reliability of the
findings.
The more recent studies of the persistence of glass fragments transferred
glass when breaking a windowpane. This should remove this source of
uncertainty.
The two earlier works demonstrate considerable foresight, however, in
that they studied phenomena, the exact relevance of which took many years
to become accepted.
©2000 CRC Press LLC
5.2.1 Early studies
The work of Pounds (personal communication, 1977) consisted of placing a
known number of glass fragments of different sizes on different types of
garments — bulky woolen sweater, woolen sweater, Tweed jacket, sweater,
and sports jacket. The size fractions were 0.1 to 0.5 mm, 0.5 to 1 mm, and
particles bigger than 1 mm. The loss of these fragments was determined by
examining the garments after a given time lapse. This study shows that glass
fragments are lost rapidly, particularly when larger than 0.5 mm. For the
bulky woolen sweater, only 8% of the fragments were recovered after 6
hours.
Brewster et al.138 have studied the retention properties of two types of
material: cotton (denim) and wool/acrylic (proportion 70:30). Glass particles
were transferred using an air rifle and counted under a microscope after 30
minutes and 1, 2, 4, and 6 hours. The results show that the percentage of
retained particles depends on the size of the fragments and that particles
between 1 and 0.5 mm in size are retained for the longest time. It is surprising
that fragments smaller than 0.5 mm are not retained longer. Plausible reasons
for this result include that there could a critical size for the retention of
fragments or that very small fragments are difficult to see under a microscope. This last hypothesis is supported by Pounds and Smalldon’s research28
on the searching of garments for glass.
Another surprising result is that glass is retained longer on denim than
on wool/acrylic. This result could also be explained either by the fact that
glass was examined with a microscope or by the method of transfer.
The disadvantage of these two studies is that glass was not actually
transferred by breaking a window. Therefore, it was necessary to study the
transfer and persistence of glass fragments on garments under more realistic
breaking conditions.
5.2.2 Persistence of glass on clothing
The latest studies on the persistence of glass were done by breaking a window and observing how many fragments were retained after a certain period
of time. Batten139 was the first to study actual persistence. He arranged for
eight windows (measuring approximately 1.50 × 0.60 m), four of which bore
a plastic coating on one surface, to be broken during a laboratory renovation.
The panes were broken by repeated blows from a hammer until most of the
glass had been removed from the frame. After either 30 or 60 minutes of
normal laboratory activity (not involving glass), hair combings and outer
clothing were collected from each subject and searched. Glass was recovered
from the hair combings of seven out of the eight subjects and from all items
of outer clothing, with the numbers ranging widely from 2 up to about 100
fragments. Comparative results will be shown later.
Hicks et al.,133 Hoefler et al.,113 and Cox et al.140-142 used the same clothing
and windowpanes as described previously in the section under transfer
©2000 CRC Press LLC
experiments; the respective distances at which the windows were broken
were 0.60 m and between 0.60 and 0.80 m. The first two studies consisted
of breaking a new window for each time interval (Hicks et al., 30 , 60, 120,
240, and 480 minutes; and Hoefler et al., 15, 30, 60, 90, 120, and 180 minutes)
during normal activity (Hicks et al.) or periods of walking (15 minutes) and
resting (10 minutes). The number of fragments present after time had elapsed
was counted. Cox et al. transferred glass by breaking a window; however,
instead of going to their normal activities, the strikers entered a box in which
they performed an activity (three activities were tested: 6 minutes of running,
20 minutes of walking, and 30 minutes of sitting). Five periods of the same
activity were performed in five wooden boxes. The boxes consisted of four
sides and a bottom, so it was possible to collect all glass that had fallen from
the garments during activity and to know the number of fragments that had
actually been transferred by addition of the collected fractions and glass
remaining at the end of the experiment. This procedure also allowed an
estimate of the influence of the degree of activity. In all three studies (in the
Cox et al. experiments these were the final fragments remaining), the fragments were recovered by shaking and counting under a microscope.
The results are summarized in Table 5.4. All three studies show that loss
appears to be a two-stage process consisting of a rapid initial loss and a
slower process that can extend over a longer period of time (see Figure 5.2).
Most of the fragments were lost during the first 30 or 60 minutes irrespective
of activity, and thereafter the rate of loss seemed to stabilize. Cox et al.
showed that the percentage of loss did not depend on the number of particles
transferred.
The size of the fragments recovered depended on the time elapsed: the
longer the elapsed time between search and breaking, the larger the proportion of small (0.2 to 0.5 mm) fragments. Cox et al. noted that during the
initial rapid loss of fragments the influence of size was more important than
in the second loss process, where the proportion of sizes appeared to remain
constant. No clear relationship between size and activity could be established, as different activities implied different time lapses. However, it seems
that the more vigorous the activity during the first 60 minutes, the smaller
the size of the recovered fragments.
Hicks et al. showed that the garments used had a bearing on the number
and size of the particles retained: the bulky sweater retained bigger and more
numerous fragments than the cotton tracksuit (see Figures 5.2 and 5.3). In
Cox et al. no difference between clothing was reported. However, this may
have been due to the similar nature of the garments used.
Cox et al. have also investigated the relationship between shape and
retention of fragments using the same classification as Underhill.135 The
authors concluded that “The persistence of glass fragments on the clothing
studied was similar for all three fragments shapes, irrespective of activity or
clothing type.”
As a practical conclusion, it has to be noted that even 8 hours after the
breakage it is possible to find as many as seven glass fragments on clothing.133
©2000 CRC Press LLC
Table 5.4
Distance
Number of Fragments Recovered on Garments
Research
60/100
120/150
30 minutes
minutes
minutes
Clothing Repeats Mean Range Mean Range Mean Range
Cox et al.a Shellsuit
4
11
1–28
top (L)
Running
Shellsuit
8
5–13
= 30
bottom (L)
minutes
Walking
Tracksuit
4
17
6–29
= 100
top (M)
minutes
Sitting
Tracksuit
31
6–74
= 150
bottom
minutes
(M)
Sweater (M)
4
33
4–51
Jeans (M)
17
6–29
Estimated Battenb
Top (L)
0
4
—
0.50–0.70 m
Bottom (L)
6
—
Top (L)
0
10
—
Bottom (L)
6
—
Top (H)
0
>100
—
Bottom (H)
14
—
0. 60 m
Hicks et al. Tracksuit
6
20 12–32
top (M)
Tracksuit
6
11
4–29
bottom
(M)
Estimated Hoefler
Sweatshirt 20/10/
4
0–9
0.50–0.70 m et al.c
(L)
5
Jeans (L)
20/10/
6
3–10
5
0. 80 m
Hicks et al. Pullover
6
10
4–17
(H)
Jeans (L)
6
12
5–25
0. 50 m
7
1–12
9
1–14
2
1–3
4
1–7
23
8–39
12
0–26
13
7–17
8
2–11
11
9
2
3
2
2
31
35
13
8–13
1–9
—
—
—
—
—
—
5–30
26
18
—
—
—
—
—
—
7
10–42
8–25
—
—
—
—
—
—
2–16
6
2–10
5
1–8
2
1–5
0
0–1
5
2–11
3
2–4
10
5–21
8
4–10
13
4–28
6
2–9
a
Cox et al. used time intervals of 30, 100, and 150 minutes.
b For each of the eight experiments, a new type of clothing was used. In order to compare times,
clothing with the same retention are presented. These clothes are the following: (30 minutes)
nylon kagoul, smooth hopsack weave, cotton T-shirt, jeans, woolen sweater, thick cord; (60
minutes) acrylic sweater, smooth weave, acrylic sweater, jeans, woolen sweater, thick cord.
c In this research the number of repeats depends on the time interval: for time intervals of 15
and 30 minutes, there were 20 repeats; 10 repeats for 60 minutes; and 5 repeats for the remaining
intervals.
Note: The retention properties are indicated as L (low), M (medium), and H (high).
5.2.3 Persistence of glass on shoes
Pounds (personal communication, 1977) reported the number of fragments
transferred to the soles of shoes. Broken window glass was placed onto a
linoleum-covered floor, and the particles were transferred to the soles by
treading on the floor. It was found that fragments on soles are lost very
rapidly. In the case of rubber shoes, only 5% of the particles were retained
©2000 CRC Press LLC
Number of fragments
140
120
100
80
60
40
20
0
0
0.5
1
2
4
8
Time (h)
Figure 5.2 Persistence of glass fragments on the breaker’s cotton sweater. A person
standing at 50 cm broke the pane with a hammer.
Number of fragments
60
40
20
0
0
0.5
1
2
4
8
Time (h)
Figure 5.3 Persistence of glass fragments on the mock accomplice’s woolen pullover. The pane was broken with a hammer with the mock accomplice standing at 80
cm.
after 30 minutes; however, deeply embedded fragments may stay for a
considerable time. The author mentions that often the only way to recover
such fragments is probe searching with a dissecting needle. When searching
a leather sole visually with a microscope after 4 minutes, 12% of the particles
were recovered, whereas 45% were recovered with a probe. For rubber soles,
the respective numbers are 7 and 12%. Unfortunately, there is no mention
of the total number of fragments recovered.
Holcroft and Shearer’s137 experiments also involved transferring glass
to shoes by walking through broken glass. The individual walked for a
further 5 minutes before the training shoes were packaged. This experiment
©2000 CRC Press LLC
was performed four times; the total number of glass fragments recovered in
each experiment was four, nine, eight, and seven, respectively.
5.2.4 Persistence of glass in hair
Batten139 studied the persistence of glass in hair. Eight windows were broken
with a hammer, and glass was searched for in the hair of the breakers (or
bystanders) after 30 and 60 minutes. It is not known if the windows were
broken by eight individuals or fewer. The results are shown in Table 5.5.
Table 5.5
Persistence of Fragments in Hair
Experiment
1
2
3
4
and
and
and
and
Number of fragments recovered Number of fragments recovered
in hair after 30 minutes
in hair after 60 minutes
5
6
7
8
9
22
3
3
0
3
4
2
As in other experiments, there is some variation. However, the study
shows that a considerable amount of glass may be recovered from the hair
of a person who has broken a window.
5.3 Main results of the studies
At the beginning of this chapter, we cited ten factors that could affect transfer
and persistence studies.
The size of the window. Results from several studies suggest that it is
the area of damage to a window, rather than the size of the window itself,
which relates to the number of fragments produced by backward fragmentation and may subsequently be retained on clothing.
The type and thickness of the glass. Limited information from the Locke
studies suggests that the harder the glass is to break, the more fragments
are produced when it does break.
How the window was broken. More fragments seem to be transferred to
the clothing of the perpetrator when a window is broken by multiple blows
than when it is broken by a single blow. It also appears that a rapid break
with a heavy weight produces fewer fragments than a slower break with a
lighter weight.
The position of the offender relative to the window. If a person is more
than 1 m from a breaking window, then it is likely that very few fragments
will be transferred to and retained on the clothing. If a person is close to a
breaking window, and the point of impact is above waist level, it is likely
©2000 CRC Press LLC
that considerably more fragments will be retained on the upper, rather than
the lower clothing.
Whether or not entry was gained to the premises. This parameter does
not seem to have a great influence on the number of fragments transferred.
The nature of the clothing worn by the suspect. This has a large effect on
the number of fragments that would be recovered. Results of the studies
suggest that there are several interrelated factors involved, including the
coarseness of the fabric and the construction of the garment.
The activities of the suspect between the times of the incident and the
arrest. There is very little reliable information on this topic, but Cox et al.
suggest that the more vigorous the activity, the more rapid the rate of loss
will be.
The length of time between the arrest and the clothing being taken. T h e
persistence experiments show that the longer the time interval, the less glass
will be remaining.
The way in which the clothing was obtained from the suspect. Va r i o u s
studies (for example, Hoefler et al. or Allen et al.) demonstrate that much
glass can be lost when clothing is removed.
The weather at the time of the incident. Damp or wet clothing would
appear to be more retentive than dry clothing.126-129
Recently, there has been much research on transfer and persistence. It is
true that there are gaps in our knowledge, but it is also true that the studies
described previously have provided us with much valuable information.
When they are considered collectively, they suggest the following general
principles.
1. Even when experimental conditions are carefully controlled, the
number of fragments produced when a window is broken can vary
considerably.
2. At least a third of all casework-sized fragments produced by backward fragmentation have original surfaces.
3. In most experiments described in the studies, glass was found on the
clothing of an individual who was close to a breaking window.
4. The main factors affecting the number of glass fragments found on an
individual exposed to a breaking glass window are the distance of the
individual from the window and the nature of the clothing worn.
5. It is likely that more fragments will be found on the footwear or socks
than on the pants of a person exposed to a breaking window when
that person was standing more than 1 m from the window.
©2000 CRC Press LLC
6. Only a very small proportion of fragments on one item are likely to
be transferred to a second item by contact between them.
7. Persistence appears to be a two-stage process. The initial loss in the
first 30 or 60 minutes is very rapid, the second process is slower, and
after 8 hours at least it is still possible to find glass on clothing.
5.4 Modeling glass transfer and making estimates
If addressing propositions at the activity level, Bayesian interpretations of
glass evidence require an estimate of Tn, “the probability that n fragments
of glass were recovered given that an unknown number were transferred
from the crime scene and retained on the offender.”
This section describes the use of simple modeling techniques as a method
for consistent and objective evaluation of the transfer probabilities.143 We
also describe how specific uncertainties can be handled.
5.4.1 Graphical models
The modeling used in this section is described in two phases. The primary
phase constructs a simple deterministic model of the transfer, persistence,
and recovery processes. This model describes the factors thought to be
involved, the parameters that characterize each factor, and the dependencies
that exist between these parameters. This primary model does not allow for
any uncertainty. The secondary phase uses this primary model to construct
a formal statistical graphical model to describe the stochastic nature of the
transfer process.
The idea to use a directed graph to represent a statistical model is not a
new one,144 but developments in the use of these ideas in Bayesian analysis
of expert systems have only come about relatively recently (see Reference
145 for a comprehensive review). We are aware that Dr. Ian Evett is using
these models in the area of fiber evidence (personal communication, 1995).
Construction of a graphical model can be divided into three distinct
stages. The first qualitative stage considers only general relationships between
the variables of interest, in terms of the relevance of one variable to another
under specified circumstances or conditions.145 This stage is equivalent to
the aforementioned primary modeling phase and leads to a graphical representation (a graphical model) of conditional independence that is not
restricted to a probabilistic interpretation. That is, the qualitative stage,
through the use of a formal graphical model, describes the dependencies
between the variables without making any attempt to describe the stochastic
nature of the variables. For example, studies have shown that the distance
of the breaker from the window influences the number of fragments that
land on the breaker’s clothing. Therefore, distance and the number of fragments that land on the suspect would be included in the graphical model.
The second quantitative stage would model the dependency between these
two variables. This probabilistic stage introduces the idea of a joint distribu-
©2000 CRC Press LLC
tion defined on the variables in the model and relates the form of this
distribution to the structure of the graph from the first stage. The final
quantitative step requires the numerical specification of the necessary conditional probability distributions.145
Use of a graphical model is appealing in the modeling of a complex
stochastic system because it allows the “experts” to concentrate on the structure of the problem before having to deal with the assessment of quantitative
issues.
A graphical model consists of two major components, nodes (representing variables) and directed edges. A directed edge between two nodes, or
variables, represents the direct influence of one variable on the other. To
avoid inconsistencies, no sequence of directed edges that return to the starting node are allowed, i.e., a graphical model must be acyclic. Nodes are
classified as either constant nodes or stochastic nodes. Constants are fixed
by the design of the study and are always founder nodes (i.e., they do not
have parents). Stochastic nodes are variables that are given a distribution
and may be children or parents (or both).146 In pictorial representations of
the graphical model, constant nodes are depicted as rectangles and stochastic
nodes are depicted as circles.
5.4.2 A graphical model for assessing transfer probabilities
The processes of transfer and persistence can be described easily, but are
difficult to model physically. The breaker breaks a window either with some
implement (a hammer or a rock) or by hand. Tiny fragments of glass may
be transferred to the breaker’s clothing. The number of fragments transferred
depends on the distance of the breaker from the window (because of the
backscatter effect demonstrated by Nelson and Revell2). The activity of the
breaker, the retention properties of the breaker’s clothing, and the time until
the breaker’s clothing is confiscated are some of the factors that determine
how many fragments will fall off the breaker’s clothing.
We have discussed the factors important to take into consideration in
order to estimate Tn. As it is unclear how to model some of these factors, the
proposed model considers only the major effects of the position, time, garment type, and the laboratory examination.
Figure 5.4 is a very simplistic graphical model that describes how distance, time, garment type, and the lab examination will affect the final
number of fragments observed on the suspect. The model can be described
thus: the number of fragments transferred to the breaker directly depends
on the distance of the breaker from the window during the breaking process.
The number of fragments that are still on the breaker’s clothing at each
successive hour up to time t depends on the number of fragments that were
initially transferred to the clothing, the number of fragments lost in the
previous hour, the time since the commission of the crime, and the glass
retention properties of the breaker’s clothing. At time t, some number of
fragments have remained on the breaker’s clothing, and how many of those
are observed depends on how many are recovered in the laboratory.
©2000 CRC Press LLC
Distance
Transfer
Time
Persistence j
Garment
type
j = 1 ...t
Lab
examination
Figure 5.4
ments.
Persistent
fragments
Observed
data
A simple graphical model for the transfer and persistence of glass frag-
Each of these steps can be resolved into more detail to specify the full
graphical model used in the transfer simulation program that provides the
results presented here. The full graphical model is described in detail in
Appendix A. The probabilistic modeling and quantitative assessment for the
full model is described in Appendix B.
5.4.3 Results
The general probability distribution of Tn is analytically intractable. However, it is possible to approximate the true distribution by simulation methods. If the parameters that represent the process are provided, then it is
possible to simulate values of n, the number of fragments recovered, by
generating thousands of random variates from the model. If enough values
of n are simulated, then a histogram of the results will provide a precise
estimate of Tn. This simulation process can be thought of as generating
thousands of cases where the crime details are approximately the same and
©2000 CRC Press LLC
observing the number of fragments recovered. Obviously, this is a computationally intensive process, and so to this end a small simulation program
has been written in C++, with a Windows 95/98/NT® interface to display
the empirical sampling distribution of n conditional on the initial information
provided by the user. As well as allowing the user to specify the initial
conditions, the program contains the full graphical model and allows the
user to manipulate the dependencies between key processes.
Figure 5.5 shows the empirical distribution function of n, given the
following information.
Figure 5.5
Empirical distribution of n.
The breaker was estimated to be 0.5 m from window. Please note that
we do not take this value as known. The breaker is at a fixed, but unfortunately unknown, distance and we take 0.5 m as an estimate. The model
incorporates uncertainty regarding this estimate into its procedure.
Given that the breaker was 0.5 m from the window when he/she broke
it, on average 120 fragments would be transferred to the breaker’s clothing.
On average the breaker would be apprehended between 1 and 2 hours after
breaking the glass.
©2000 CRC Press LLC
In the first hour, the breaker would lose, on average, 80 to 90% of the
glass transferred to his clothing and, on average, 45 to 70% of the glass
remaining on his clothing in each successive hour until apprehension.
Figure 5.6 shows the model with the same assumptions, but does not
allow the distance of the breaker from the window to affect the mean number
of fragments transferred to the breaker.
Figure 5.6
Empirical distribution of n without the distance effect.
Estimation of transfer probabilities is a formidable task for the glass
examiner and, in fact, may be overwhelming. This is because of the inherent
variability of the process, the multiple factors affecting transfer and retention,
and the fact that many pieces of information may be lacking in the known
case circumstances. However, it is seldom that nothing is known. For
instance, the retentive properties of the clothing are usually estimable (since
the clothing has been submitted). We argued earlier that even if there is no
knowledge regarding the probable distance from the window of the breaker,
people adopt only a few positions when breaking a window. In some cases
there is no information regarding the time lapse between breakage, and
packaging of the clothing. Even in this circumstance we can safely assume
that it is almost impossible to package clothing less than 1 hour after the
©2000 CRC Press LLC
breakage, and for glass examiner purposes 30 hours approximates infinity.
The graphical model gives a method by which many of these uncertainties
may be accommodated.
For a very wide range of values in the parameters, the probability of
recovering any given number of fragments is found to be low. However, it
must be borne in mind that we are actually interested in the ratio of this
probability to the probability of this number of fragments recovered from
the clothing of a person unrelated to a crime. In many circumstances the
probability calculated under the assumption of transfer, persistence, and
recovery will, although low, be many times higher than the chance of this
number of fragments recovered from the clothing of a person unrelated to
a crime.
The effect of time is very marked: the longer the time until apprehension,
the more fragments that are lost.
The effect on the distribution of the number of fragments recovered, by
allowing the distance of the breaker from window to affect the average
number of fragments transferred, is quite interesting. The distance factor is
specified as a stochastic node in the full graphical model (see Appendix A).
The assumed distribution of distance puts high probability on the true distance being close to the estimated distance. However, the true distance can
be either less than or greater than the estimated distance. If the true distance
is less than the estimated distance, the mean number of fragments that can
be transferred to the breaker is increased. If the true distance is greater than
the estimated distance, the mean number of fragments that can be transferred
to the breaker is decreased. Because it is more likely (in this model) that the
true distance will be less than or equal to the estimated distance, the overall
mean number of fragments transferred in the simulations is higher than for
an experiment where the true distance is fixed. This increase is represented
by the longer tail of the empirical distribution function for the number of
fragments recovered, given in Figure 5.5, as opposed to that in Figure 5.6.
The conclusions are robust to the naive distributional assumptions made
in Appendix B. The graphical modeling technique offers consistency and
reliability unparalleled by any other method. The answers it provides may
be refined with more knowledge and better distributions, but that such naive
assumptions can provide such a realistic distribution function (J. Buckleton
and J.R. Almirall, personal communication, 1996) suggests that it is the
dependencies that drive the answer and not just the distributional assumptions. A full Bayesian approach would take the graphical model as an informative prior distribution and modify it by casework data to obtain the
posterior distribution.
The previous model was experimental, and with hindsight some of the
modeling assumptions may have been naive. For instance, we draw attention
to the assumed distribution regarding distance: di ~ Gamma(d). The scientist
estimates that the breaker was d meters from the window. Allowing the true
value, di, to be a Gamma distributed random variable represents the fact that
©2000 CRC Press LLC
the true distance is more likely to be closer than further away. It seems likely
that this assumption may need research.
The performance of the model, however, may be sufficiently startling
that it may be plausible to utilize it in casework immediately.
The authors wish to encourage experimental work on these distributional assumptions.
5.4.4 Conclusions from the modeling experiment
We seek to make three main points. First, that forensic scientists should be
aware of what question they are answering when they assess the value of
the Tn terms in a Bayesian interpretation. We suggest that this question be
“what is the probability of recovering n fragments from a suspect’s clothing
given that: (1) an unknown number of fragments were transferred to the
suspect from the crime scene, (2) something is known about the retention
properties of the suspect’s clothing, (3) the distance of the breaker from the
window has been estimated, and (4) the time between the commission of
the crime and the arrest has been estimated.”
The second point is that regardless of the method used to evaluate these
probabilities, with the exception of T0, the estimates should be low. That is,
there is a very small chance of recovering any given number of glass fragments from a suspect even if he/she is caught immediately.
The third and most important point is that simple graphical models
combined with extensive simulation effectively model the current state of
knowledge on transfer and persistence problems. The graphical modeling
technique offers consistency and reliability unparalleled by any other
method and, thus, must be recommended as the method for estimating
transfer probabilities.
The software used in this book is available for Windows 95/98 or higher
only by e-mailing the authors.*
5.5 Appendix A — The full graphical model for assessing
transfer probabilities
Figure 5.7 is a more complete graphical model of the transfer and persistence
processes. This model can be interpreted as follows.
• The breaker is a fixed, but unknown distance, di, from the window.
An estimate of this distance, d, is made by the forensic scientist.
• At this distance, on average λi fragments are transferred to the breaker’s clothing during the breaking process. The average, λi, depends
on an estimated average, λ from experimental work. An unknown
number of fragments, x0, are actually transferred.
* E-mail: [email protected].
©2000 CRC Press LLC
Figure 5.7
ments.
A formal graphical model of the transfer and persistence of glass frag-
• Of the x0, on average 100 × q% will become stuck in the pockets or
cuffs or seams of the clothing or in the weave of the fabric. An unknown proportion, q0, of fragments are actually in this category. The
persistence of these fragments is modeled separately because they
have a higher probability of remaining on the clothing.
• The breaker is not apprehended until an unknown number of hours,
–
t, later. t is estimated by t .
©2000 CRC Press LLC
• During the first hour, on average 100 × p0% of the x0 and 100 × p0*%
of the q0 fragments initially transferred are lost, where p0 and p0* are
unknown, but lie somewhere on the intervals defined by [l0, u0] and
[l0*, u0*], respectively. b0 and b0* are the actual number of fragments lost.
• In each successive hour, on average 100 × pi% of the xj and 100 × pi*%
of the qj fragments remaining from the previous lost, are lost, where
pi and pi* are unknown, but lie somewhere on the intervals defined
by [li, ui] and [li*, ui*], respectively. bi and bi* are the actual number of
fragments lost.
• At the end of t hours there are a total of yi fragments remaining. On
average, 100 × R% of these are recovered by the forensic scientist,
where R is unknown, but lies somewhere on the interval defined by
[lR, uR]. b is the actual number of fragments not recovered.
• Finally, Y fragments are observed.
5.6 Appendix B — Probabilistic modeling and quantitative
assessment
As noted in Section 5.4, specification of a graphical model consists of three
stages. Now that the first stage is complete, the probabilistic stage can be
dealt with. The beauty of the graphical model comes from its conditional
independence properties. It can be shown that the distribution of a child
node depends only on the distribution of its parents.145,146 This implies that
if one distribution does not model a variable very well, then a better distribution may be substituted and the resulting changes will be automatically
propagated through the affected parts of the model. In that vein the following
distributions are proposed for each of the variables.
di ~ Gamma(d): The scientist estimates that the breaker was d meters from
the window. Allowing the true value, di, to be a Gamma distributed
random variable represents the fact that the true distance is more likely
to be closer than further away.
wi ~ Normal(1, 0.25); (l i = (l wi ~ Normal(l , l /4)): The net result of letting λi
be a normal random variable is a more spread out distribution for the
number of fragments actually transferred.
x0 ~ Poisson (e(1-di/d)l i): The weight e(1–di/d) adjusts the mean number of fragments transferred on the basis of the true distance. If di < d, then the
breaker is closer to the window than estimated, and, therefore, a higher
number of fragments will be transferred on average. Similarly, if di > d,
then the breaker is further from the window than estimated, and, thus,
a smaller number of fragments will be transferred on average. This
spread out Poisson distribution accurately reproduces the results given
in Hicks et al.133
©2000 CRC Press LLC
q0 ~ Binomial(x0, q): On average, 100 × q% of the fragments will get stuck
in the pockets or cuffs or seams of the clothing or in the weave of the
fabric.
x1 = max(x0 – q0 – b0,0); b0 ~ Binomial(x0, p0); p0 ~ Uniform[l0, u0]:
q1 = max(q0 – b0*, 0); b0* ~ Binomial(q0, p0*); p0* ~ Uniform[l0*, u0*]:
Letting p0(pi*) be uniform represents the uncertainty over the number of
fragments actually lost in the first hour and the retention properties of
the garment. The number of fragments lost in the first hour is significantly higher than those lost in successive hours. Hicks et al.133 show
that the larger fragments are lost very quickly.
t ~ Nbin(n, r): The number of hours until apprehension, t, is modeled as a
negative binomial random variable. The negative binomial distribution
models the number of coin flips that are needed until n tails are observed,
–
where the probability of a head is r. If n = t = lt + ut/2, and r = 0.5, where
lt and ut are estimated lower and upper bounds on the time, then the
negative binomial provides a good estimate for the distribution of t, with
–
a peak at t and a tail out to the right. This says that the time between
commission of the crime and arrest is more likely to be shorter than
estimated rather than longer.
xj + 1 = max(xj – bj, 0); bj ~ Binomial(xj, pi); pi ~ Uniform[li, ui]:
qj + 1 = max(qj – bj*, 0); bj* ~ Binomial(qj, pi*); pi* ~ Uniform[li*, ui*]:
In each successive hour the suspect will lose, on average, 100 × pi% of
the fragments that remain on his clothing, where pi is a uniform random
variable that can take on values between li and ui. The number of fragments actually lost in the first hour is b, where b is a binomial random
variable with parameters xi and pi.
yi = qt + xt; b ~ Binomial(yi, 1 –R); R ~ Uniform[lR, uR]; Y = max(yi – b, 0):
Finally, after t hours the suspect is apprehended; his clothing is confiscated and examined, and of the yi fragments that remain on his clothing,
b fragments are not found following the results of Pounds.28
©2000 CRC Press LLC
chapter six
Statistical tools and software
The theory presented in the preceding chapters requires the examiner to
make expert assessments of various probabilities and to evaluate the “relative rarity” of glass. In order to do this, a body of survey data is required.
However, such collections are often available. Which collections are most
suitable has been discussed in more detail in Chapter 4. We propose in this
chapter to discuss what to do with the data; how to estimate the relative
frequency of a given glass; and, in particular, how to construct histograms
and density estimates. Several computer programs, which have been developed to assist the glass examiner when comparing measurements and assessing the value of glass in general, will also be discussed.
6.1 Data analysis
6.1.1 Histograms and lookup tables
We introduce a histogram discussion as a prelude to discussing density
estimates. We take this long route because most people are unfamiliar with
the concept of density estimation and find it a foreign idea. However, it is
such a powerful tool in forensic interpretation that considerable effort is
warranted.
A histogram is a statistical graph that displays frequency information.
It is instructive to see how a histogram is constructed before discussing its
relative merits.
6.1.1.1 Constructing a histogram
In the following method it is assumed that we have a set of N RI measurements, where N is a moderately large number (N = 200 is reasonable). We
denote each measurement xi for i = 1,…, N.
Find the minimum RI,
xmin = min xi
i
©2000 CRC Press LLC
and the maximum RI,
xmax = max xi
i
Mark these values on your horizontal axis.
Divide the interval between the minimum and the maximum into a
number of equal width class intervals or bins. The number of bins, k,
is an arbitrary or ad hoc decision.
Construct a lookup table with the bins as the categories.
Place each observation into the appropriate interval in the lookup table
Count the number of observations in each class interval and draw a bar
with height proportional to that count.
It should be obvious that this is a task better suited for a computer,
especially for a large data set. The question of choosing the number of bins
is a controversial one. In order to carry out this procedure correctly, we need
to consider what properties we would like our histogram to have.
Desirable Properties of an RI Histogram
1. An accurate representation of the
distribution of RIs
2. Robust to the addition of small amounts
of new data
3. Sensitive to the addition of large amounts
of new data
4. Sufficient resolution to provide accurate
assessment of relative rarity
It should be apparent that the shape of a histogram is entirely dependent
on the number of bins. For example, the 1994 New Zealand database of
casework samples consists of 2656 measurements. From Figure 6.1 we can
see that when we use ten bins we get the basic shape of the distribution of
the data, but a fairly crude measure about the relative rarity of different RIs.
That is, glass fragments are categorized as rare, common, or very common.
While this categorization works, it is unlikely that any practicing caseworker
would find it satisfactory. When the number of bins is increased to 50, the
distribution becomes more discriminating. That is, it can distinguish between
samples of glass with about a 0.07 difference in mean (that is equivalent to
a difference of 0.00002 in RI). It is worthwhile noting that there seem to be
two outlying samples. While it is necessary to describe this part of the data,
we would really like to describe the bulk of the data better than the outlying
points. In order to do this we need to use a specialized (nonparametric data
driven) method for locating the upper and lower bounds that describe a
large portion of the data. One way would be to construct a 95% confidence
interval around the mean of the data.
©2000 CRC Press LLC
2500
2000
k=50
k=200
0
500
1000
1500
k=10
Figure 6.1
New Zealand casework data histogram with 10, 50, and 200 bins.
However, confidence intervals are poor tools for this purpose and
depend on parametric assumptions that are unwise with complex distributions such as glass frequency distributions (they work best with simple
distributions such as the normal distribution). A more robust choice is to
find the whiskers of a box and whisker plot (more commonly known as a
boxplot). The steps for constructing the whiskers for a box and whisker plot
are the following.
1. Sort the data into ascending order, so that x(i) is the ith value in the
sorted data set and x(1) ≤ x(2) ≤…≤ x(n).
2. Find the lower quartile (LO) and the upper quartile (UQ). These are
the kth smallest and kth largest observations in the data set, where
 n
floor  + 1
 2
k=
2
(6.1)
3. Find the interquartile range (IQR),
IQR = UQ − LQ
(6.2)
4. Find the approximate lower and upper bounds. These are given by
LB = LQ − 1.5 × IQR
©2000 CRC Press LLC
(6.3)
UB = UQ + 1.5 × IQR
(6.4)
This may seem difficult, but it is best illustrated by an example. Our
New Zealand data contains n = 2656 measurements. Therefore,
 2656 
floor
 +1
 2 
1328 + 1
k=
=
= 664.5
2
2
(6.5)
The fact that k is fractional means we have to take the average of the
664th and 665th smallest values for the lower quartile and the average of
the 664th and 665th largest values for the upper quartile:
LQ =
x( 664 ) + x( 665 ) 1.51593 + 1.51593
=
= 1.51593
2
2
(6.6)
and
UQ =
x( 2656−664 ) + x( 2656−665 )
2
=
x(1992 ) + x(1991)
2
=
1.51964 + 1.51964
2
(6.7)
= 1.51964.
Therefore, the interquartile range is IQR = UQ – LQ = 1.51964 – 1.51593
= 0.00371. Now, we calculate the approximate positions of the whiskers.
LB = LQ − 1.5 × IQR = 1.51593 − 1.5 × 0.00371 = 1.510365
(6.8)
UB = UQ − 1.5 × IQR = 1.51964 − 1.5 × 0.00371 = 1.525205
(6.9)
Next, we increase the lower bound until it falls on the first observation
greater than LB. For our data set this value is 1.51086. Finally, we decrease
the upper bound to the first observation less than UB. For our data set this
is 1.52518. About 95% of our data lies between these two bounds. If we use
50 bins, there are approximately 10 bins that describe this 95%. That is, the
bulk of cases will fall into one of ten categories. This is probably still too low
for most practitioners. When we increase the number of bins to 200, this
©2000 CRC Press LLC
Table 6.1 Suggested Number of Bins for New
Zealand Casework Data
Method
Sturges147
Dixon and Kronmal149
Scott150
Scott (corrected)150
Freedman and Diaconis151
Freedman and Diaconis152
Number of bins (k)
12
34
64
130
68
138
number becomes 39. We have close to the discrimination we desire without
too much loss of information.
As can be seen from the previous paragraphs, selecting the number of
bins for a histogram is not an easy task. The examiner should not be drawn
down the line of assuming that 0.0001 in RI units is anything more than
some arbitrary choice based upon “human” convenience. There are a number
of suggested automatic methods for choosing the number of bins (see Table
6.1). Sturges147 suggested that k = floor[1 + log2(N)], when N is a power of
2. Hoaglin et al.148 note that this rule can be extended to N that are not a
power of 2, but this is more for those who want an aesthetic picture rather
than to convey information. Dixon and Kronmal149 suggested that one choose
k = ceiling[10log10(N)] as an upper bound for the number of bins. Hoaglin
et al.148 note that this rule works moderately well for 20 ≤ N ≤ 300. David
Scott of Rice University in Texas, a well-known researcher in the area of
density estimation, constructs a method for an optimal bin width, based on
the assumption that the data is unimodal and symmetric.150 If this is the case,
he suggests a bin width of hN = 3.49sN –1/3, where s is the sample standard
deviation of the data set. However, this method is unlikely to provide sufficient resolution if the data is strongly bimodal or multimodal. In such cases
the data is likely to be oversmoothed. Scott150 suggests that hN be multiplied
by a correction of about a half. Freedman and Diaconis151 suggest a bin width
based on the maximum difference between the histogram and the true distribution, giving
 ln N 
hN = 1.66 s

 N 
1/ 3
(6.10)
In a second publication, based on different criteria, Freedman and
Diaconis152 suggest
hN =
2 IQR
N 1/ 3
(6.11)
Let us use our New Zealand data set to examine these formulae. Recall
N = 2656.
©2000 CRC Press LLC
400
300
200
100
0
1.50
1.52
1.54
1.56
x
Figure 6.2
New Zealand casework data histogram with 135 bins.
It is clear from our previous investigation that the choice of 12 or even
34 bins is going to be too small. When we choose k to be 135, there are 26
bins for the bulk of the data.
From Figure 6.2 it seems that 135 bins provide a nice amount of discrimination — the bins have a width of 0.00055, i.e., they will give different values
for samples with a difference in their means of 0.00055. This is about ten
times the median standard deviation for recovered samples (perhaps a little
large, but a nice compromise). The data are well described, i.e., they are
neither too clumped together nor overdispersed.
The construction of a histogram in this way is a useful way to display
the relative rarity of glass at different RIs. However, the estimation of a
frequency for a particular type of glass is better performed by a “floating
window.”
6.1.2 Constructing a floating window
Most laboratories undertaking glass examination currently report a frequency for the glass found. This is actually quite a difficult topic, and given
the marked preference of the authors for the continuous approach it will
only be discussed briefly. In Chapter 2, Section 2.3 we described coincidence
probabilities, which are the closest published formal definition of a frequency.
©2000 CRC Press LLC
We mentioned earlier that given a precise definition of what is meant
by frequency, it is easy to develop an algorithm that can estimate this probability. However, no completely satisfactory definition of “frequency” has
ever been given.
In Chapter 2 we defined a frequency (coincidence probability) as an
answer to the question: “What is the probability that a group of m fragments
taken from the clothing of a person also taken at random from the population
of persons unconnected with this crime would have a mean RI within a
match window of the recovered mean?” Defining the match window is also
difficult, and most attempts are based on applying the match criterion to m
fragments from random clothing (or other population) and the n fragments
from the control. Such a match window would be centered on the recovered
mean and would have a width determined by the match criterion.
This is the standard application of the floating window. That is a window
of width determined by the match criterion centered on the recovered mean.
This differs from the standard application of a histogram, which shows
“relative frequency” and gives an overall view of the data but does not make
an attempt to define the vague term “frequency.”
One algorithm is given in Evett and Lambert.91 We give another algorithm in Appendix A.
6.1.3 Estimating low frequencies
There is at least one major flaw in the floating window approach. What
happens when we encounter a sample of glass with a mean RI that falls
either outside the range of RI histogram or into a window with zero count,
i.e., a bin where no RIs of similar value have been seen before? It seems that
we have something of a paradox. The glass sample we have recovered is
very rare, but we cannot quantify this so that it will fit into our Bayesian
framework, i.e., we would be dividing the numerator by zero. We could take
some of the approaches used in the analysis of DNA data. For VNTR data,
the band frequencies are (arbitrarily and incorrectly) obtained by dividing
the allelic ladder up into bins so that there are at least five observations in
each bin. This method is known as rebinning. When a histogram is constructed using unequal bin widths, the area of the bar associated with the
bin represents the relative frequency of observations falling into that bin.
This is in contrast to our previous histograms where the height of the bar
alone gave the relative frequency. This, however, was really just a side effect
of having all the bars of equal width, so the area of the bars in our previous
histograms would give us the relative frequency. Rebinning would work and
is certainly conservative, in that it would give much higher frequencies for
many RIs. However, it is not an optimal approach, and there are better
methods, as we will explain shortly. Another approach, used in the analysis
of DNA, is to put the case sample into the database. That is, if we have N
measurements in our database, and a case sample with two fragments with
©2000 CRC Press LLC
an RI that has not been seen before, we would estimate the frequency of the
RI as
f=
2
N+2
While this approach is at least robust and slightly easier to justify, it is still
not optimal because it fails to recognize that RI measurements are continuous
measurements. That is, there are no gaps between the possible RI values;
any gaps observed are merely a function of the limits of accuracy in any
measurement tool. The best way to display continuous data is to construct
a density estimate.
6.1.4 Density estimates
In order to discuss density estimates we need some more terminology. Most
of it will be familiar in some form or another, or at least logical. The reader
may skip this section if he/she is familiar with the concept of continuous
random variables and probability density functions.
6.1.4.1 Random variables and probability density functions
Definition
If X is a variable that records the result of a random experiment, then X is
said to be a random variable.
Example. If X records the number of heads in ten tosses of a fair coin,
then X is a random variable
Definition
If X is a random variable with a countable number of possible values or gaps
between the possible values, then X is said to be a discrete random variable.
Example. In the previous example X can take on any value in the set
{0, 1, 2,…, 8, 9, 10}. In this example we can count all 11 possible outcomes.
There are no outcomes in between any of those we have listed. Therefore,
X is a discrete random variable.
Definition
If X is a discrete random variable, then the probability that X takes on a
particular value, x, is given by Pr(X = x). Pr(X = x) is called the probability
function of X.
Example. Assuming that each toss of the coin has no effect on the
following coin tosses and that the only two outcomes on each toss are heads
©2000 CRC Press LLC
or tails, X is said to be a binomial random variable or have a binomial
distribution, with parameters n = 10 and p = 0.5. The probability density
function for a binomial random variable is given by
 n
n− x
Pr(X = x) =   p x (1 − p)
 x
(6.12)
For example, if n = 10 and p = 0.5, then the probability of getting exactly five
heads on ten throws is given by
 10
0.5
Pr(X = 5) =   0.5 5 (1 − 0.5) ≈ 0.246
 5
(6.13)
i.e., about 25%.
Note that Pr(X = x) is often written in shorthand as Pr(x).
Definition
If X is a random variable with an uncountable number of possible outcomes
or there are no gaps between the possible outcomes, then X is said to be a
continuous random variable.
Example. If X is the RI of a piece of glass, then X is a continuous
random variable.
Definition
If X is a continuous random variable, and the probability that X lies between
two points a and b is given by
Pr( a ≤ X ≤ b) =
b
∫ f (t).dt
(6.14)
a
then f(x) is said to be a probability density function (pdf) for X.
Note that it is important to note that for a continuous random variable
with pdf f(x), f(x) ≠ Pr(X = x). In fact, for a continuous random variable, Pr(X
= x) = 0. This effectively says that if we could measure RI to infinite precision,
the chance of observing two RIs that were exactly equal is zero.
Given that RI is a continuous random variable, we would like to find
the pdf for RI. However, as with most physical models, the answer is not
analytically tractable. What we must do instead is try to elicit the density
function from our data. One way of thinking about this is to go back to our
coin tossing model. We are going to toss a coin ten times and observe the
number of heads. We would like to be able to specify the probability of
observing any one of the possible outcomes. We have seen that we can work
©2000 CRC Press LLC
Properties of Probability Density Functions
1.
2.
0 ≤ Pr(a ≤ X ≤ b) ≤ 1
Pr( X ≥ a) = Pr( X > a) and Pr( X ≤ b) = Pr( X < b)
so
Pr(a ≤ X ≤ b) = Pr(a < X ≤ b) = Pr(a ≤ X < b) = Pr(a < X < b)
3.
4.
*
1 − Pr( X > a) = Pr( X ≤ a)
Pr(φ) =
∫
a
a
f (t ).dt = 0, Pr(Ω) =
∞
∫ f (t).dt = 1
−∞
*This is not true for a probability function.
this out using the binomial distribution. But what would we do if we did
not know about the binomial distribution? One way is to repeat the experiment many times and record the outcomes. Using the frequency of the
outcomes, we could construct a histogram for the outcomes. If we could
repeat this experiment often enough (in fact, forever), then eventually we
would come up with a histogram that would have almost the same shape
as the probability function for a binomial random variable. Because the
histogram makes no assumptions about the true distribution of the outcomes, it is a nonparametric density estimate of the probability function.
Now, let us think about our RI distribution. If we could measure a large
number of RIs and put them into a histogram, we would have a nonparametric density estimate of pdf for RI. However, as we put more and more
data into the histogram, we would like the histogram to become more and
more accurate. In order to do this we would need to increase the number of
bins. That means that the bin widths would get smaller and smaller because
the range of RI is between approximately 1.48 and 1.56. If this process could
be repeated forever, then eventually the bin widths would be zero and we
would be left with a nice smooth curve — i.e., the limiting histogram is the
probability density function.
This is best illustrated with an example. Assume Z is a standard normal
random variable. That is Z ~ N(0,1), then the pdf for Z is
f ( z) ~
1
 1 
exp − z 2 
 2 
2π
(6.15)
Figure 6.3 demonstrates our point. As we take bigger and bigger samples
from the normal distribution and increase the number of bins, the resulting
histogram becomes closer and closer to the probability density function for
the normal distribution.
So it would seem we have a method for finding the pdf for RIs. However,
it is obvious that we cannot collect an infinite amount of glass and determine
©2000 CRC Press LLC
N = 100, k = 10
N = 1,000, k = 50
N = 10,000, k = 100
N = 50,000, k = 500
N = 100,000, k = 1,000
N = 500,000, k = 1,000
Figure 6.3 The limiting histogram of samples from a normal distribution. N is the
sample size and k is the number of class divisions.
©2000 CRC Press LLC
its RI. Currently, an experienced operator with a GRIM II can measure, at
most, about 50 fragments in an 8-hour period. What we need is some way
of smoothing our histogram or getting a smoothed histogram without needing all that data.
In order to do this we must relax one of our assumptions — the assumption about knowing nothing under the underlying distribution of the data.
We know from the exploration of our case data that there are a number of
modes (or bumps) in the data set and that the data is fairly symmetric around
a couple of large central modes. We might think then that a localized assumption of normality is okay. That is, within small “windows” the RI of a piece
of glass is a normally distributed random variable. So finally we get into
density estimation.
6.1.5 Kernel density estimators
A traditional kernel density estimate works at smoothing the data by replacing each data point with a probability density function (typically a normal),
or kernel, centered at the data point. The resulting density estimate at any
point, x, is the mean of the values of the individual densities for each datum
ˆ is conat that point. The smoothness of the resulting density estimate, f(x),
ˆ
trolled by a tuning parameter or “window” or “bandwidth,” h. f(x) is defined
as
f̂ ( x ) =
1
Nh
N
∑ K 
i =1
x − xi 

h 
(6.16)
where h is the window width, and the smooth kernel function,
K (x) =
1 −
e
2π
x2
2
is the probability density function for a standard normal variable.
The astute reader should be asking at this point, “Where does h come
from? Isn’t choosing h just like choosing the number of bins for a histogram?”
The simple answer, of course, is yes. However, a kernel density estimate
with a well-chosen tuning parameter is often much more robust than a
histogram. The reason for this is the nature of the kernel function. The kernel
function essentially acts as a weighting function. Single datum with extreme
values will have much less weight than a large group of data with similar
values.
6.1.5.1 What is a good tuning parameter for a kernel density estimator?
Just as with choosing the number of bins for a histogram, there is no single
or simple answer to this question. Scott153 proves that if the data are truly
©2000 CRC Press LLC
Figure 6.4
data.
h/4
h/2
h
2h
The effect of h on a kernel density estimate of the New Zealand casework
normally distributed, then choosing h = 1.06sxN–1/5, where sx is the sample
standard deviation of the data, is optimal. However, we know that our data
most definitely is not normal, and so the criteria for choosing h will be similar
to that of choosing the number of bins.
Figure 6.4 shows the effect of changing the bandwidth on the kernel
density estimate. If h gets too small, the resulting density estimate is jagged
and has a lot of variation. When data is added to this density estimate it will
register with stronger effect than if a larger h had been used. When h is too
large, for example, when we choose a parameter that is twice as large as the
one suggested by Scott,153 too much detail is obscured. In the plot shown in
Figure 6.4 the outlying data is not visible in the density estimate using 2h.
Figure 6.4 suggests that an h of somewhere between 3/4 h and 11/4 h would
produce a reasonable density.
6.2 Calculating densities by hand
In order to implement our recommended approach (the continuous Bayesian
approach), two densities are required. We need to know (1) the probability
©2000 CRC Press LLC
of getting the analyzed characteristics of the unknown and the control knowing that they come from the same origin and (2) the probability of getting
the analyzed characteristics of the unknown and the control knowing that
they come from different origins. The density in the numerator (1) is a value
from a t-distribution with either integral (equal variance t) or nonintegral
(Welch’s modification) degrees of freedom. Computer programs provide
these values, but if such programs are not available, then they may be
obtained “by hand” using a spreadsheet like Microsoft Excel which is
intended to show readers how to obtain these values.
The second density that is required comes from survey data and appears
in the denominator. It is also intended to demonstrate how that may be
obtained by hand. Imagine that we proceed with an example where there is
one recovered group and one control group. The analytical data are shown
in Table 6.2.
Table 6.2 Summary Data for a Control and
Recovered Group of Glass
N
Sample mean
Sample SD
Control
6
1.51804
0.000055
Recovered
4
1.51804
0.000149
The mean of the recovered group is 1.51804. Therefore, in the denominator, we require the probability density of glass on clothing at 1.51804.
Table 6.3
RI
Glass Frequency Data Excerpts from New Zealand Glass Surveys
Vehicle Vehicle
Vehicle
Window pre-1983 1984–1987 1988–1992 Container Patterned
1.5176
2
1.5177
4
1.5178
3
1.5179
2
1.5180
3
1.5181
1
1.5182
2
1.5183
2
1.5184
1
Total samples 400
17
14
11
5
11
10
19
36
24
507
9
4
5
0
2
2
11
16
16
219
5
1
2
0
0
0
2
2
4
183
0
0
2
0
0
0
0
2
2
74
1
1
0
0
1
1
1
2
0
128
Flat
1
3
3
2
2
2
1
0
1
272
Earlier we discussed the values shown in Table 6.3 and stated that we
would prefer glass on clothing surveys in most instances. We proceed here
as if the reader does not have available a glass on clothing survey. Let us
imagine that we can concentrate on the vehicle pre-1983 survey. We are only
doing this to simplify the problem at this stage. We cannot think of many
scientific reasons to concentrate on this survey other than it appears to be
the most “common” at this stage. We require the probability density at
©2000 CRC Press LLC
0.0217
1.51804 for pre-1983 vehicles. There are 11 observations out of 507 in this
histogram bar. Consider this histogram bar:
The area of the bar is the probability (0.0217). The width is 0.0001 RI
units. The probability density is the height. This is 0.0217/0.0001 = 217.
A related approach might be to smooth the distribution first. This would
“take out” local lumps and hollows. There are sophisticated and effective
ways to do this, but a rough (but workable) method is simply to smooth
over three (or five) histogram bars. In this case, smoothing over three bars
we get a probability of 5 + 11 + 10/507 and a width of 0.0003 RI units, giving
a density of 171.
In the absence of a clothing survey the analyst is faced with something
of a dilemma. Which survey should be used? This has troubled many examiners, but is in fact more a “mental” problem than a real one. The analyst
appears to have four choices:
1. Use a clothing survey.
2. Weight surveys of the above type according to subjective estimates
of the likely source of glass on clothing (predominantly tableware).
3. Choose the most conservative option.
4. Determine the source (or sources) by elemental analysis or other
method.
Such methods are not “state-of-the-art” statistics, but they may get laboratories “going” in a workable way.
The density for the numerator remains to be calculated by hand. Recall
the statistics that were produced.
t-test: The test statistic is 0.05 for a t-distribution on 8 degrees of freedom
Welch’s t: The test statistic is 0.04 for a t-distribution on 3.57 degrees of
freedom.
©2000 CRC Press LLC
Table 6.4 Probability Density from the tDistribution
Test statistic
d.f. = 8
d.f. = 3.57
0.00
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
0.09
0.10
0.500
0.496
0.492
0.488
0.485
0.481
0.477
0.473
0.469
0.465
0.461
0.500
0.496
0.493
0.489
0.485
0.482
0.478
0.474
0.471
0.467
0.463
Note: The d.f. = 8 and d.f. = 3.57 columns give
the probability mass to the right of the test
statistic for 8 and 3.57 degrees of freedom,
respectively.
Table 6.4 shows a section of a Microsoft Excel table produced using the
TDIST function.
For the Welch distribution we require the density at a test statistic of
0.04. This is approximately (0.485–0.482)/(0.05–0.04), that is 0.003/0.01 = 0.3,
which unfortunately is not on an RI scale. To rescale we observe that
V=
(x − y)
2
s x2 s y
+
n m
(6.17)
therefore, a test statistic (V) of 1 is equivalent to a difference on the means of
2
s x2 s y
+
n m
which in this case was 7.78 × 10–5. To rescale we observe that the width of
the “bar” is not 0.01, but 0.01 × 7.78 × 10–5 = 7.78 × 10–7 and therefore the
density is 0.003/7.78 × 10–7 = 3856. This is the density to compare with the
“conservative” density of about 217 from the denominator.
6.3 Computer programs
No single comprehensive set of programs exists at this time that implements
our recommended approach to casework. This is a serious flaw given the
computational complexity of some of our recommended procedures. However, the components of a workable system exist in various places, and this
section reviews these and suggests how they might be cobbled together.
©2000 CRC Press LLC
6.3.1 The Fragment Data System (FDS)
This program was developed in the mid-1990s by Richard Pinchin and Steve
Knight in order to compare RI measurements. The FDS program takes data
either by manual input or directly from GRIM II. It can display this data in
an elegant graphical form and perform objective grouping algorithms (Evett
and Lambert, personal communication, 1991). The graphical output and
ability to color and tag that data can be of great assistance to subjective
grouping decisions. Once the fragments have been grouped, this program
enables us to perform Student’s t-test, the Welch test, and the continuous
LR (using nonmatching glass from the LSH survey).104
This program can interact with CAGE (Computerized Assistance for
Glass Evidence) for Windows in a limited fashion and thereby permit implementation of the continuous approach.
6.3.2 STAG
STAG (STatistical Analysis of Glass) was written by James Curran as part of
his Ph.D. thesis in 1996 with suggestions and bug testing from John Buckleton, Kevan Walsh, Sally Coulson, and Tony Gummer from the Institute of
Environmental Science Ltd., as well as Chris Triggs from the University of
Auckland in New Zealand.
STAG revolves around sample data entered by hand into a spreadsheetlike interface. The user has the ability to designate samples as control and
recovered samples, indicate where groupings might exist in samples, and
store case notes on each sample. The data is entered in temperature form
and converted into RI using a calibration line that each laboratory may set.
STAG incorporates many of the procedures and tests discussed in James
Curran’s thesis. In particular it is the only program to include the grouping
algorithm of Triggs et al.77,78 The manual or automatic grouping information
is included in the subsequent analyses.
STAG has limited support for the Bayesian approach — it can produce
continuous LRs based on the user’s own glass database. STAG is available
free from James Curran (e-mail: [email protected]).
6.3.3 Elementary
Elementary, written by James Curran, is a tool for the analysis of elemental
concentration data (e.g., output from ICP-MS, ICP-AES, etc.). Elemental
provides a simple interface for the implementation of Hotelling’s T2, as
described in Reference 154 and the implementation of the continuous LR as
described in Reference 155. It is a crude tool in that it requires the user to
have (1) their own database and (2) input files organized by another program
(such as a text editor or Microsoft Excel) in a format described in the help
file. Elementary is available free from James Curran (e-mail: [email protected]). Both Elementary and STAG are Microsoft Win-
©2000 CRC Press LLC
dows programs written in C++ for the Win32 API. They have not been ported
to other platforms, although the essential elements of both programs are
fairly transportable.
6.3.4 CAGE
CAGE is written in CRYSTAL, a report management language. The underlying program uses a Rule Based Forward Chaining logic that works with
the glass examiner to focus on relevant questions and to assist in producing
an interpretation and associated paperwork. Written by Richard Pinchin and
John Buckleton at the Home Office Central Research Establishment, Aldermaston, England (as it then was) in 1988, it is now considered by both authors
to be in need of modernization despite being upgraded to the Windows 95
format and partially incorporating the continuous approach. Currently, this
program has only been used in routine casework in New Zealand.
The program incorporated the approach of Evett and Buckleton7 in a
form intended for casework use. The formula is generalized and can handle
most combinations of control and recovered groups. Data was taken from
the McQuillan and Edgar survey.156 After selecting an item, the user can
choose several menus (see a description of the program or a table of terms),
alter default data, and do a full case (with specific or average probabilities)
or a precase analysis. CAGE assists the user when assessing transfer probabilities either by showing a summary of research or by giving the transfer
probabilities assessed by other experts in similar cases (the similarity of the
case is based on the type of control, the distance between offender and
window, time elapsed, retention properties, and the type of glass). This
program also offers the glass examiner very useful “help pages.”
In the late 1990s CAGE was still relevant in that it produced (in symbolic
form) the equations for any combination of controls and recovereds and
attempted to explain each term. In addition, it performed the small services
of providing the data in usable form from the ME survey and of performing
the arithmetic. Minor upgrades, that could be done, are to introduce propositions management (CAGE enables us to address only two propositions);
to put the menu preassessment first and have safeguards to warranty that
transfer probabilities are first estimated and then fragments searched. Much
research was performed in the late 1990s and so an upgrade would involve
presenting these new results and integrating external modules such as the
program Transfer. Accessibility on the World Wide Web could entice more
people to use CAGE and perhaps would create a bigger database of RI data.
Currently, FDS and CAGE are available for purchase from the U.K.
Forensic Science Service.
6.4 Summary
In this chapter we have shown how to use statistical tools in order to estimate
1/f or the continuous LR. We have presented computer programs and meth-
©2000 CRC Press LLC
ods that enable us to compare measurements and to assist the forensic
scientist when assessing the value of glass. The aim of our last chapter will
be to present how we would report results in a glass case. We will see how
it is possible to avoid fallacies, particularly what has been called the prosecutor’s fallacy. We will continue with the presentation of four examples
involving single and multiple controls and recovered, using elemental analysis.
6.5 Appendix A
If matching is performed using the Student’s t-test, then a window is set as
Recovered mean ± Tcrit S p
1 1
+
n m
(6.18)
where Tcrit is the critical value of the T statistic. If Welch’s statistic is used
then the window is given by
Recovered mean ± Vcrit
2
sx2 sy
+
n m
(6.19)
A frequency is then calculated either by counting the number of observations that fall into this window or by summing areas after smoothing.
There is a logical flaw in this approach that we are unable to resolve.
The value of the control standard deviation appears in the definition of the
width of the window. This is because it is integral to the definition of the
match criteria. However, strictly speaking, an answer to the question —
“what is the probability that a group of m fragments taken from the clothing
of a person also taken at random from the population of persons unconnected with this crime would have a mean RI within a match window of
the recovered mean?” — has nothing to do with the control glass sample.
Hence, it is difficult to see why the control standard deviation should appear
anywhere in the definition. This difficulty appears to be associated with the
general difficulty in defining a frequency and is solved by the continuous
approach.
©2000 CRC Press LLC
chapter seven
Reporting glass evidence
Reporting is a source of constant difficulty in forensic science.157 The challenge is to reduce the information created by the examination and typically
held in a considerable case file to a few pages, or even a few sentences, of
text. This, by its very nature, involves information loss. To make it slightly
harder we ask that this be done without the use of formal probabilistic
symbols and preferably in plain language. Anyone who for a moment thinks
that this task in any way is easy should perform the following experiment.
Get a colleague to choose a statement on a topic that
you are not particularly familiar with. Sit in one seat
for an hour listening to a boring radio program then
ask your colleague to read the statement to you once.
Ask your colleague to put the inflections and pauses
in the wrong places because this is what will assuredly
happen when a statement is read to the court. Challenge yourself to see what level of understanding and
recall you can achieve.
The aim of this discussion is to attempt to identify those aspects of glass
evidence most pertinent to the juries’ understanding of the evidence and to
preserve those aspects in the report. In the ensuing discussion we follow
strongly the thinking of Evett.158
Interpretation of evidence takes place within a framework of
circumstances.
Because the scientist’s interpretation depends upon the circumstances,
it would seem reasonable for these to be made explicit in the statements.
To interpret evidence it is necessary to consider at least two
propositions.
©2000 CRC Press LLC
We hold this to be a basic tenet of interpretation; therefore, we suggest
that these two propositions should be explained in the statement.
It is necessary for the scientist to consider the probability of
the evidence given each of these propositions.
The results of the consideration may be reported with or without numbers or with both numbers and a verbal explanation. If numbers are used,
we recommend something along the lines:
“This evidence is approximately (or at least depending upon
approach) x times more likely if the clothing was close to the
window at the pharmacy when it broke than if it has never
been near this window.”
The same statement can be made without the use of numbers. Various
scales have been offered, and we reproduce one here in Table 7.1 (from
Reference 158).
Table 7.1
LRs
A Verbal Scale for Reporting
LR
1 to 10
10 to 100
100 to 1000
1000 upwards
Verbal equivalent
Limited support
Moderate support
Strong support
Very strong support
Therefore, an interpretation giving an LR of approximately 500 might be
reported as
In my opinion this evidence strongly supports the proposition that this clothing was close to the window at the pharmacy when it broke.
This scale is symmetric. That is that LRs less than 1 should be reported
as supporting the alternative hypothesis. This scale is not confined to glass
evidence alone and is, in fact, perfectly general.
Table 7.2 illustrates how arbitrary these qualitative scales can be.
In addition to the arbitrary nature of these scales, confusion may arise
over the verbal description of numbers. This is best illustrated by noting that
the British interpret the word “billion” to mean 1012, while the U.S. interprets
the word as 109.
In general, we would advocate that when reporting an LR greater than
one, the value of the LR should be included with the verbal equivalent. For
example, the interpretation giving an LR of approximately 500 might be
reported as
©2000 CRC Press LLC
Table 7.2 Qualitative Scale for Reporting the Value of the Support of
the Evidence for C against
Verbal Equivalent
Slight increase in support
Increase in support
Great increase in support
Very great increase in support
LR
LR Approx.
1 to 100.5
100.5 to 101.5
101.5 to 102.5
102.5 upwards
1 to 3.15
3.15 to 31.5
31.5 to 315
315 upward
From Aitken, C. G. G., Statistics and the Evaluation of evidence for Forensic Scientists,
John Wiley & Sons, New York, 1995, p. 52. With permission.
This evidence is approximately 500 times more likely if the
clothing was close to the window at the pharmacy when it
broke than if it has never been near this window. In my
opinion this evidence strongly supports the proposition that
this clothing was close to the window at the pharmacy when
it broke.
In order to have something concrete upon which to focus discussion, we
return to the simplest type of case.
Example 7.1 One group, one control
The suspect was apprehended 30 minutes after the breakage. The breakage was believed to have been performed by hitting the window with a club.
Subsequently, the offender entered the premises through the broken window.
The offender is believed to have exited through a door at the rear of the
premises. Items taken or ransacked were not in the broken glass zone. Only
one perpetrator was suspected.
As a matter of discipline, it is desirable to consider certain aspects of the
case before the search is performed. Specifically, it is beneficial to consider
the probability of transfer prior to knowing how many fragments were
potentially transferred. This should increase the objectivity in these assessments. Determining the transfer probabilities presents difficulties for most
glass examiners. However, the experienced glass examiner probably knows
more than she or he realizes, and some training and thought experiments,
some of which were given in Chapter 5, quickly demonstrate that the examiner is in a better position to attempt this admittedly difficult task than the
jury.
From the work that has been done on glass transfer and persistence
(Chapter 5), it appears reasonable to model the probability distribution for
the number of fragments recovered after a given time by the method of
Curran et al.143
Figure 7.1 shows the probability that m fragments were recovered from
the suspect given the following information:
• The breaker was estimated to be 0.5 m from window.
©2000 CRC Press LLC
Figure 7.1
An empirical distribution function of m.
• Given that the breaker was 0.5 m from the window when he broke
it, on average, 120 fragments would be transferred to the breaker’s
clothing.
• On average, the breaker would be apprehended between 0 and 1h
after breaking the glass.
• In the first hour, the breaker would lose, on average, 80 to 90% of the
glass transferred to his clothing and, on average, 45 to 70% of the
glass remaining on his clothing in each successive hour until apprehension.
Using this information we estimate T0 = 0.079 and T4 = 0.042
A forensic examination is now performed. Four fragments were recovered from a full search of the clothing (T-shirt and jeans), including the
pockets. Control and recovered measurements were as follows:
Recovered: 1.51820, 1.51812, 1.51798, and 1.51787
Control: 1.51805, 1.51809, 1.51794, 1.51801, 1.51805, and 1.51809
Summary statistics for the control and recovered measurements are
shown in Table 7.3. A test using Welch’s modification to the t-test appears
©2000 CRC Press LLC
Table 7.3 Summary Statistics for a Control and
Recovered Sample of Glass
N
Sample mean
Sample SD
Control
Recovered
6
1.51804
0.000055
4
1.51804
0.000149
appropriate in view of the fact that the recovered glass is from clothing.
Using this test, the glass matches at the 1% level.
t-test: The test statistic is 0.05. The 1% critical value for a t-distribution
on 8 degrees of freedom is t8 (0.005) = 3.35.
Welch’s t: The test statistic is 0.04. The 1% critical value for a t-distribution on 3.57 degrees of freedom is t3.57 (0.005) = 5.01.
The LR is given by
P(E|C , I )
(
P E|C , I
)
= T0 +
P0 .Tn
P1 .Sn . f
(7.1)
From the LSH survey, P0 = 0.25, P1 = 0.22, and S4 = 0.02. Determining f is
not straightforward. Remember the difficulty we had in phrasing the question to form this probability (previously termed the coincidence probability).
We seek to answer a question of the form: “If I find a group of six fragments
on the clothing of a person unconnected with the smashing of the window
in this case, what is the probability that it would be the same as the recovered
glass?” The “same” is the difficult part (a precise solution exists in the
continuous approach which was introduced in Chapter 3), but here we define
it as glass within a match window of the recovered mean. The match window
is defined by the nature of the comparison test, in this case Welch’s t. This
is actually quite complicated. Here, we will assume that a rough estimate is
available as follows.
The Welch test defines a match window* by
y ± tψ (0.005)
2
sx2 sy
+
n m
(7.2)
where tψ(0.005) is the (two-tailed) 1% critical value of the t-distribution with
degrees of freedom given by
* The presence of the terms Sx and n in this equation is a result of the difficulty in defining the
frequency. Strictly, we are considering where the glass did not come from the control; therefore,
it is clear that these parameters should not appear in the formulation. The resulting formula
given here is a best attempt to shoehorn the confidence approach into the correct thinking.
©2000 CRC Press LLC
ψ=
 s 2 sy2 
x
 + 
 n m
2
(7.3)


sy4
sx4
+
 2

2
 n (n − 1) m (m − 1) 
For this case, this formula defines a window centered on 1.51804 with
lower and upper bounds given by subtracting and adding 0.00022, respectively. This interval contains all of the 1.5179, 1.5180, 1.5181, and 1.5182
histogram bars and 0.1 of the 1.5183 bar (containing the information from
1.51825 to 1.51835) and 0.3 of the 1.5177 bar. Using this approximate method,
we obtain (eventually) f ≈ 0.04.
Some (many) laboratories do not have a “clothing” survey, so we offer
here an approximate and most probably conservative method. However, we
strongly recommend the use of clothing surveys.
Suppose data of the type given in Table 7.4 have been collected. From
this data we can obtain a set of frequency estimates by dividing each count
by the number of samples and, in the case of the vehicles that change quickly
with time, weighting according to modern vehicle numbers.
Table 7.4
RI
1.5176
1.5177
1.5178
1.5179
1.5180
1.5181
1.5182
1.5183
1.5184
Glass Frequency Data Excerpts from New Zealand Glass Surveys
Vehicle
Vehicle
Vehicle
Window pre-1983 1984–1987 1988–1992 Container Patterned
2
4
3
2
3
1
2
2
1
17
14
11
5
11
10
19
36
24
9
4
5
0
2
2
11
16
16
5
1
2
0
0
0
2
2
4
0
0
2
0
0
0
0
2
2
1
1
0
0
1
1
1
2
0
Flat
1
3
3
2
2
2
1
0
1
Table 7.5 Summary the Glass
Frequency Data Excerpts from
New Zealand Glass Surveys
Type
Frequencies
Window
Vehicle
Container
Patterned
Flat
2.3%
7.2%
2.4%
1.9%
2.5%
None of these frequencies in Table 7.5 are exact, nor are they a substitute
for a well-constructed clothing survey. However, they do give the range of
most probable answers, and since we believe that most glass on clothing is
©2000 CRC Press LLC
container glass, it is not inconsistent with the 4% answer from the clothing
survey. This type of survey, while inferior, is easily updated without a complete rework of the whole survey. For instance, in New Zealand the vehicle
data is reworked every 3 years by adding a new block of data and reweighting. Window glass surveys separated by over 10 years in New Zealand were
very similar and they are not updated as frequently.
Using the previous estimates, the LR becomes
LR = 0.079 +
0.25 × 0.042
0.22 × 0.02 × 0.04
(7.4)
= 59.73 ≈ 60
We propose here to investigate this example further. First, we will try to
verbalize this answer. This is an issue of great difficulty and importance.
Second, we will investigate aspects of this approach such as sensitivity of
this answer to some of the data estimates and to search procedures.
7.1 Verbalization of a likelihood ratio answer
Strictly, a phrasing of this answer could be the following: “This evidence is
about 50 times more likely if the clothing was within two meters of the
window when it broke than if it has never been near this window.” In saying
this, we have done several things. First, we have rounded down the answer,
which reflects our own view of the precision (single digit) possible with this
type of analysis. Second, we have avoided the fallacy of the transposed
conditional. At this time we make the point that this evidence does not imply
that “it is 50 times more likely that the clothing was within 2 meters of the
window when it broke than if it has never been near this window.” This is
a transposed statement and is false. It is also noted that, following forensic
practice, we have avoided the words guilt and innocence and have substituted hypotheses.
C:
– The clothing was within 2 m of the window when it broke.
C : The clothing has never been near this window.
While we advocate placing before the court a numerical estimate of the
value of the evidence as given previously, many examiners do not feel that
this is appropriate.
Typical LRs for glass evidence lie in the range of 0.1 to 1000. It is difficult
to get much below 0.1, which tends to occur when no glass is found but
transfer was reasonably likely. Equally, very high LRs, above 1000, happen
when a lot of matching glass is found or when two or more controls have
matching glass. The scale effectively “runs out of words.”
We concede that verbalizing an LR is more difficult than merely “giving
a frequency.” The very concept of an LR is unfamiliar to most juries, although
©2000 CRC Press LLC
it seems likely that Bayesian logic is very akin to the natural thinking patterns
of most people. These verbal difficulties are part of the barrier to widespread
use of LRs. Here, we merely claim that the use of this approach effectively
incorporates information about the presence of glass on the clothing per se
and that it offers a logical way in which to view glass (and in effect all)
evidence. Those persons offering a “frequency” will very shortly get into
just as much verbal trouble when they try to either define their frequency
or to incorporate data about how rare large groups of glass are on a person’s
clothing. However, such difficulties are useful in that they encourage discussion and questioning of the evidence and the mode of presentation.
7.2 Sensitivity of the likelihood ratio answer to some of the
data estimates
Each of the terms in the LR is an estimate. None of them is known exactly.
As such, they have error (sampling or other) in them. Some of these are
particularly important in their effect on the LR and some are almost unimportant. In Example 7.1, T0 is unimportant (it is important whenever a control
exists, but there is no matching glass for this control). Again, in this case P0,
P1, and f are reasonably well supported by data, and while they may have
sampling error, this effect is unlikely to be great in comparison to the very
difficult terms T4 and S4.
When we consider S4, we find that the difficulty exists in two places.
First, S4 is small, and estimating small things is hard (ask the DNA analysts).
The reason for this is quite straightforward. If S4 is in the order of 0.02 (or
less), then we only expect to see it twice in a survey of 100 sets of clothing.
Realistically, we might see it a few more times or not at all because of
sampling error. It is even more difficult to estimate the higher terms. In a
typical case it is not unusual for us to require S10 which is very small in the
LSH survey for nonmatching glass. Estimating this term with any accuracy
is very difficult.
There is one, at least, strong counterargument. Many of the terms are
well supported by data. For instance, S1, S2, and S3 appear to be reasonably
well estimated. This does not leave much for the other terms (since they
must add to 1) so that even if we do not know exactly what they are, we do
know that they are small. In addition, the distribution is expected to be
continuous; that is, there is no logical reason to expect S10 to suddenly be a
peak if S9 and S11 are small. In fact, it seems reasonable to expect S10 to be
smaller than S9, but larger than S11. This suggests that judgment might be
reasonably used to estimate these terms. If S1 is about 70%, S2 about 16%,
and S3 about 6%, we have only 8% of the area to distribute among the
remaining possibilities. This logic certainly suggests that S10 is less than 1%.
These terms could be examined by standard sampling error type techniques, and the most obvious is the bootstrap.159 The clothing survey (LSH
in this case) could be resampled and the relevant terms reestimated many
©2000 CRC Press LLC
times, producing an estimate of the variance in the LR produced by sampling
error. We are unaware of any work done along these lines, and in view of
the subjective nature of many of the estimates this may not be warranted.
The second issue arises when there are, say, 100 fresh glass fragments
in the debris. The analyst examines ten of these and finds that they all match.
Should statements be made about the 10 or the 100? We can certainly guarantee that ten match, but it seems likely that many more do. The difference
in the LR equations is quite extreme:
LR = T0 +
P0T10
P1S10 f
or
(7.5)
LR = T0 +
P0T100
P1S100 f
which clearly would give very different answers. It is safe in this hypothetical
case to use
LR = T0 +
P0 T10
P1S10 f
since this is almost certainly the numerically smaller (and hence more conservative) answer.
7.3 The effect of search procedures
Imagine that we had done the previous case differently. We had examined
the T-shirt and found the four matching fragments and then stopped. The
formula for the LR is still
LR = T0 +
P0 T4
P1S4 f
(7.6)
but we need to consider exactly what the P, S, and T terms mean.
Pi is the probability of finding i groups of glass under this search procedure. Therefore, in the first example it was under a search of the T-shirt
and jeans including pockets. In the second instance it is under a search of
the T-shirt only. Glass on the surface of upper clothing is rarer than if the
whole set of clothes and the pockets are searched; therefore, we expect the
P terms to change, reflecting less glass.
Si is the probability of a group of size i under this search procedure.
Ti is the probability that i fragments were found using this search procedure given that an arbitrary number were transferred and had persisted.
©2000 CRC Press LLC
To demonstrate this effect we take P0, P1, and S4 from the LSH survey
of the surfaces of garments only and leave the other terms constant. These
are 0.42, 0.26, and 0.01, respectively, giving an LR of 165, which is higher
than the 59 achieved by a complete search. Part of this difference is artificial
and comes from the loss of information in summarizing the glass as coming
from the clothing rather than specifying each position. This was a simplification that we made to facilitate an already complex problem. However,
much of the difference is real and suggests that if a large group of matching
glass (three or more fragments) is found on the upper clothing or hair, then
the search should be stopped.
7.4 Fallacy of the transposed conditional
This term relates to a common logical fallacy also termed the prosecutor’s
fallacy.160 This has been well written about94,102,161 and will only be referred
to briefly here. It has been more of an issue in DNA than anywhere else but
is, in fact, entirely general.
Let us imagine that the glass evidence warrants an LR of, say, 500 (the
frequentist approach is just as susceptible to this fallacy). Strictly, this means
–
that the evidence is 500 times more likely if C is true than if C is true. This
is a statement about the probability of the evidence, E. The fallacy is to
–
interpret it as a statement about the probability of C (or C ). In words, we
(correctly) state that the evidence is 500 times more likely if Mr. Henry broke
the window than if he did not. We might erroneously state, or be misconstrued as stating, that it is 500 times more likely that Mr. Henry broke the
window.
It is very easy to see this in symbols. The correct expressions say something about Pr(E|C), the probability of the evidence given a hypothesis. The
incorrect expressions state something about Pr(C) or Pr(C|E), the probability
of a hypothesis. The name “fallacy of the transposed conditional” comes
from equating Pr(E|C) with Pr(C|E). The vertical bar is the conditioning
bar, and the terms have been “transposed” about it.
A commonly used analogy goes as follows:
The probability that an animal has four legs if it is a cow is
very high.
This statement is not the same as
The probability that an animal is a cow if it has four legs is
very high.
The first statement is true and the second is obviously false. This example
might seem to suggest that spotting correct and incorrect statements is easy.
It is not.
©2000 CRC Press LLC
The following are some forensic examples from Evett.162
The probability of finding this blood type if the stain had
come from someone other than Mr. Smith is 1 in 1000.
This statement is correct. The event is “finding this blood type” and the
conditioning information is that it came from some other person. The condition is made clear by the use of “if.”
The probability that the blood came from someone other than
Mr. Smith is 1 in 1000.
This statement is wrong. It is the most common form of the transposed
–
conditional. It is the spoken equivalent of Pr(C |E) = 1/1000; the probability
of a hypothesis given the evidence rather than the other way around.
Both of these statements are reasonably easily assigned as correct or
false. However, some are much more difficult:
The chance of a man other than Mr. Smith leaving blood of
this type is 1 in 1000.
This statement can be read two ways, and a good discussion can usually ensue.
Guidelines for Avoiding the Transposed Conditional (After Evett)
1. It is inadvisable to speculate on the truth of a hypothesis without
considering at least one other hypothesis.
2. Clearly state the alternative hypotheses that are being considered.
3. If a statement is to be made of probability or odds, then it is good practice
to use “if” or “given” explicitly to clarify the conditioning information.
4. Do not offer a probability for the truth of a hypothesis.
We drew several conclusions from this discussion. First, it teaches us
some guidelines that might keep us from committing this fallacy. These are
enumerated below and are well discussed in Evett.162 However, first it seems
worthwhile observing that the frequency with which lawyers, judges, some
forensic scientists, and newspapers make this transposition suggests that
–
they want to, understandably, make some statement about C or C. The
observation that such statements are desirable does not in itself show how
they can be made. We observe, however, that Bayes Theorem shows in a
simple manner how knowledge regarding Pr(E|C) can be used to make
statements about Pr(C|E). We cannot imagine any simpler proof of the need
for Bayesian reasoning in the evaluation of evidence. We challenge any
person using the frequentist type reasoning to demonstrate any similar simple way in which evidence may be interpreted. Certainly none has been
forthcoming.
©2000 CRC Press LLC
References
1. Marris, N. A., Interpreting glass splinters, Analyst, 59, 686–687, 1934.
2. Nelson, D. F. and Revell, B. C., Backward fragmentation from breaking glass,
J. Forensic Sci. Soc., 7, 58–61, 1967.
3. Ojena, S. M. and De Forest, P. R., Precise refractive index determination by
the immersion method, using phase contrast microscopy and the mettler hot
stage, J. Forensic Sci. Soc., 12, 312–329, 1972.
4. Evett, I. W., The interpretation of refractive index measurements, Forensic Sci.,
9, 209–217, 1977.
5. Lindley, D. V., A problem in forensic science, Biometrika, 64, 207–213, 1977.
6. Evett, I. W., A Bayesian approach to the problem of interpreting glass evidence
in forensic science casework, J. Forensic Sci. Soc., 26, 3–18, 1986.
7. Evett, I. W. and Buckleton, J. S., The interpretation of glass evidence: a practical
approach, J. Forensic Sci. Soc., 30, 215–223, 1990.
8. Walsh, K. A. J., Buckleton, J. S., and Triggs, C. M., A practical example of glass
interpretation, Sci. Justice, 36, 213–218, 1996.
9. Zerwick, C., A Short History of Glass, Harry N Adams Publishers, New York,
1990.
10. Tooley, F. V., The Handbook of Glass Manufacture, Vol. II, 3rd ed., Ashlee Publishing Co. Inc., New York, 1984.
11. Sanger, D. G., Identification of toughened glass using polarized-light, J. Forensic Sci. Soc., 13, 29–32, 1973.
12. Glathart, J. L. and Preston, F. W., The behaviour of glass under impact: theoretical considerations, Glass Technol., 9, 89–100, 1968.
13. Preston, F. W., A study of the rupture of glass, J. Soc. Glass Technol., 10, 234–269,
1926.
14. Preston, F. W., A study of the rupture of glass, J. Soc. Glass Technol., 10, 234–269,
1926.
15. Murgatroyd, J. B., The significance of surface marks on fractured glass, J. Soc.
Glass Technol., 26, 156, 1942.
16. Miller, E. T., Forensic glass comparisons, in Forensic Science Handbook, Vol. 1,
Saferstein, R., Ed., Prentice-Hall, Englewood Cliffs, NJ, 1982.
17. Oughton, C. D., Analysis of glass fractures, Glass Ind., 26, 72–74, 1945.
18. Renshaw, G. D. and Clarke, P. D. B., Variation in thickness of toughened glass
from car windows, J. Forensic Sci. Soc., 14, 311–317, 1974.
19. Buckleton, J. S., Axon, B. W., and Walsh, K. A. J., A survey of refractive index,
color and thickness of window glass from vehicles in New Zealand, Forensic
Sci. Int., 32, 161–170, 1986.
20. Slater, D. P. and Fong, W., Density, refractive index and dispersion in the
examination of glass: their relative worth as proof, J. Forensic Sci., 27, 474–483,
1982.
21. Greene, R. S. and Burd, D. Q., Headlight glass as evidence, J. Crim. Law
Criminol. Police Sci., 40, 85, 1949.
22. Fong, W., A thermal apparatus for developing a sensitive, stable and linear
density gradient, J. Forensic Sci. Soc., 11, 267–272, 1971.
23. Beveridge, A. D. and Semen, C., Glass density method using a calculating
digital density meter, J. Can. Forensic Sci., 12, 113–116, 1979.
©2000 CRC Press LLC
24. Gamble, L., Burd, D. Q., and Kirk, P. L., Glass fragments as evidence: a
comparative study of physical properties, J. Crim. Law Criminol., 33, 416–421,
1943.
25. Dabbs, M. D. G. and Pearson, E. F., Some physical properties of a large number
of window glass specimens, J. Forensic Sci., 17, 70–78, 1972.
26. Stoney, D. A. and Thornton, J. I., The forensic significance of the correlation
of density and refractive index in glass evidence, Forensic Sci. Int., 29, 147–157,
1985.
27. Thornton, J. I., Langhauser, C., and Kahane, D., Correlation of glass density
and refractive index-implications to density gradient construction, J. Forensic
Sci., 29, 711–713, 1984.
28. Pounds, C. A. and Smalldon, K. W. The Efficiency of Searching for Glass on
Clothing and the Persistence of Glass on Clothing and Shoes, Forensic Science
Service, 1977.
29. Fuller, N. A., Some Observations on the Recovery of Glass by Hair Combing,
Metropolitan Police Forensic Science Laboratory, Aldermaston, London, England, 1996.
30. Locke, J. and Scranage, J. K., Breaking of flat glass, part 3: surface particles
from windows, Forensic Sci. Int., 57, 73–80, 1992.
31. Elliot, B. R., Goodwin, D. G., Hamer, P. S., Hayes, P. M., Underhill, M., and
Locke, J., The microscopical examination of glass surfaces, J. Forensic Sci. Soc.,
25, 459–471, 1985.
32. Locke, J. and Zoro, J. A., The examination of glass particles using the interference objective. 1. Methods for rapid sample handling, Forensic Sci. Int., 22,
221–230, 1983.
33. Locke, J. and Elliott, B. R., The examination of glass particles using the interference objective. 2. A survey of flat and curved surfaces, Forensic Sci. Int., 26,
53–66, 1984.
34. Lloyd, J. B. F., Fluorescence spectrometry in the identification and discrimination of float and other surfaces on window glasses, J. Forensic Sci., 26,
325–342, 1981.
35. Locke, J., A simple microscope illuminator for detecting the fluorescence of
float glass surfaces, Microscope, 35, 151–157, 1987.
36. Zernike, F., Phase contrast, a new method for the microscopic examination of
transparent objects, Physica, 9, 686–693, 1942.
37. Zoro, J. A., Locke, J., Day, R. S., Bademus, O., and Perryman, A. C., An
investigation of refractive index anomalies at the surfaces of glass objects and
windows, Forensic Sci. Int., 39, 127–141, 1988.
38. Locke, J., Underhill, M., Russell, P., Cox, P., and Perryman, A. C., The evidential value of dispersion in the examination of glass, Forensic Sci. Int., 32,
217–219, 1986.
39. Cassista, A. R. and Sandercock, P. M. L., Effects of annealing on toughened
and non-toughened glass, J. Can. Soc. Forensic Sci., 27, 171–177, 1994.
40. Miller, E. T., A rapid method for the comparison of glass fragments, J. Forensic
Sci., 10, 272–281, 1965.
41. Koons, R., Peters, C., and Rebbert, P., Comparison of refractive index, energy
dispersive x-ray fluorescence and inductively coupled plasma atomic emission spectrometry for forensic characterization of sheet glass fragments, J.
Anal. Atomic Spectrom., 6, 451–456, 1991.
©2000 CRC Press LLC
42. Underhill, M., Multiple refractive index in float glass, J. Forensic Sci. Soc., 22,
169–176, 1980.
43. Davies, M. M., Dudley, R. J., and Smalldon, K. W., An investigation of bulk
and surface refractive indices of flat window glasses, patterned window glasses, and windscreen glasses, Forensic Sci. Int., 16, 125–137, 1985.
44. Locke, J., Sanger, D. G.;, and Roopnarine, G., The identification of toughened
glass by annealing, Forensic Sci. Int., 20, 295–301, 1982.
45. Locke, J. and Hayes, C. A., Refractive index measurements across glass objects
and the influence of annealing, Forensic Sci. Int., 26, 147–157, 1984.
46. Underhill, M., The Annealing of Glass. Notes for Caseworkers, Metropolitan
Police Forensic Science Laboratory, Aldermaston, London, England, 1992.
47. Locke, J. and Hayes, C. A., Refractive index measurements across a windscreen, Forensic Sci. Int., 26, 147–157, 1984.
48. Locke, J. and Rockett, L. A., The application of annealing to improve the
discrimination between glasses, Forensic Sci. Int., 29, 237–245, 1985.
49. Winstanley, R. and Rydeard, C., Concepts of annealing applied to small glass
fragments, Forensic Sci. Int., 29, 1–10, 1985.
50. Marcouiller, J. M., A revised glass annealing method to distinguish glass types,
J. Forensic Sci., 35, 554–559, 1990.
51. Reeve, V., Mathieson, J., and Fong, W., Elemental analysis by energy dispersive x-ray: a significant factor in the forensic analysis of glass, J. Forensic Sci.,
21, 291–306, 1976.
52. Terry, K. W., Van Riessen, A., Lynch, B. F., and Vowles, D. J., Quantitative
analysis of glasses used within Australia, Forensic Sci. Int., 25, 19–34, 1984.
53. Ryland, S. G., Sheet or container? — Forensic glass comparisons with an
emphasis on source classification, J. Forensic Sci., 31, 1314–1329, 1986.
54. Howden, C. R., Dudley, R. J., and Smalldon, K. W., The analysis of small glass
fragments using energy dispersive X-ray fluorescence spectrometry, J. Forensic
Sci. Soc., 18, 99–112, 1978.
55. Howden, C. R., The Characterisation of Glass Fragments in Forensic Science
with Particular Reference to Trace Element Analysis, University of Strathclyde, Glasgow, 1981.
56. Keeley, R. H. and Christofides, S., Classification of small glass fragments by
X-ray microanalysis with the SEM and a small XRF spectrometer, Scanning
Electron Microsc., I, 459–464, 1979.
57. Rindby, A. and Nilsson, G., Report on the xrf µ-Beam Spectrometer at the
Swedish National Forensic Science Laboratory with Special Emphasis on the
Analysis of Glass Fragments, Linköping University, Forensic Science Center,
1995.
58. Coleman, R. F. and Goode, G. C., Comparison of glass fragments by neutron
activation analysis, J. Radioanal. Chem., 15, 367–388, 1973.
59. Hickman, D. A., A classification scheme for glass, Forensic Sci. Int., 17, 265–281,
1981.
60. Almirall, J. R., Cole, M., Furton, K. G., and Gettinby, G., Discrimination of
glass sources using elemental composition and refractive index: development
of predictive models, Sci. Justice, 38, 93–101, 1998.
61. Buscaglia, J. A., Elemental analysis of small glass fragments in forensic science,
Anal. Chim. Acta, 288, 17–24, 1994.
©2000 CRC Press LLC
62. Hickman, D. A., Elemental analysis and the discrimination of sheet glass
samples, Forensic Sci. Int., 23, 213–223, 1983.
63. Hickman, D. A., Glass types identified by chemical analysis, Forensic Sci. Int.,
33, 23–46, 1987.
64. Andrasko, J. and Maehly, A. C., Discrimination between samples of window
glass by combining physical and chemical techniques, J. Forensic Sci., 23,
250–262, 1978.
65. Catterick, T. and Hickman, D. A., Sequential multi-element analysis of small
fragments of glass by atomic-emission spectrometry using an inductively
coupled radiofrequency argon plasma source, Analyst, 104, 516–524, 1979.
66. Koons, R., Fiedler, C., and Rawalt, R., Classification and discrimination of
sheet and container glasses by inductively coupled plasma-atomic emission
spectrometry and pattern recognition, J. Forensic Sci., 33, 49–67, 1988.
67. Wolnik, K., Gaston, C., and Fricke, F., Analysis of glass in product tampering
investigations by inductively coupled plasma atomic emission spectrometry
with a hydrofluoric acid resistant torch, J. Anal. Atomic Spectrom., 4, 27–31,
1989.
68. Catterick, T. and Hickman, D. A., The quantitative analysis of glass by inductively coupled plasma-atomic emission spectrometry: a five element survey,
Forensic Sci. Int., 17, 253–263, 1981.
69. Catterick, T., Hickman, D. A., and Sadler, P. A., Glass glassification and discrimination by elemental analysis, J. Forensic Sci. Soc., 24, 350–350, 1984.
70. Hickman, D. A., Harbottle, G., and Sayre, E. V., The selection of the best
elemental variables for the classification of class samples, Forensic Sci. Int., 23,
189–212, 1983.
71. Houk, R. S., Mass spectrometry of inductively coupled plasmas, Anal. Chem.,
58, 97A–105A, 1986.
72. Jarvis, K. E., Gray, A. L., and Houk, R. S., Handbook of Inductively Coupled
Plasma Mass Spectrometry, Blackie & Son Ltd., Glasgow and London, 1992.
73. Zurhaar, A. and Mullings, L., Characterization of forensic glass samples using
inductively coupled plasma mass spectrometry, J. Anal. Atomic Spectrom., 5,
611–617, 1990.
74. Parouchais, T., Warner, I. M., Palmer, L. T., and Kobus, H. J., The analysis of
small glass fragments using inductively coupled plasma mass spectrometry,
J. Forensic Sci., 41, 351–360, 1996.
75. Evett, I. W. and Lambert, J. A., The interpretation of refractive index measurements III, Forensic Sci. Int., 20, 237–245, 1982.
76. Curran, J. M., Triggs, C. M., Buckleton, J. S., and Coulson, S., Combining a
continuous Bayesian approach with grouping information, Forensic Sci. Int.,
91, 181–196, 1998.
77. Triggs, C. M. and Curran, J. M., A Divisive Approach to the Grouping Problem
in Forensic Glass Analysis, University of Auckland, Department of Statistics,
Auckland, New Zealand, 1995.
78. Triggs, C. M., Curran, J. M., Buckleton, J. S., and Walsh, K. A. J., The grouping
problem in forensic glass analysis: a divisive approach, Forensic Sci. Int., 85,
1–14, 1997.
79. Evett, I. W., The interpretation of refractive index measurements II, Forensic
Sci. Int., 12, 37–47, 1978.
©2000 CRC Press LLC
80. Scott, A. J. and Knott, M. A., Cluster analysis method for grouping means in
the analysis of variance, Biometrics, 30, 507–512, 1974.
81. Curran, J. M., Buckleton, J. S., and Triggs, C. M., The robustness of a continuous likelihood approach to Bayesian analysis of forensic glass evidence,
Forensic Sci. Int., 104, 93–100, 1999.
82. Welch, B. L., The significance of the difference between two means when the
population variances are unequal, Biometrika, 29, 350–362, 1937.
83. Hill, G. W., Algorithm 395: student's t-distribution, Commun. ACM, 13,
617–619, 1970.
84. Best, D. C. and Rayner, J. C. W., Welch’s approximate solution for the BehrensFisher problem, Technometrics, 29, 205–210, 1987.
85. Curran, J. M. and Triggs, C. M., The t-test: standard or Welch — which one
should we use?, unpublished.
86. Almirall, J. R., Characterization of Glass Evidence by the Statistical Analysis
of Their Inductively Coupled Plasma/Atomic Emission Spectroscopy and
Refractive Index Data, Nashville, TN, 1996, pp. 1–50.
87. Stoecklein, W., Determination of Source and Characterization of Glass of
International Origin, San Antonio, TX, 1996.
88. Johnson, R. A. and Wichern, D. A., Applied Multivariate Analysis, Prentice-Hall,
Englewood Cliffs, NJ, 1982.
89. Seber, G. A. F., Multivariate Observations, John Wiley & Sons, New York, 1984.
90. Stoney, D. A., Evaluation of associative evidence: choosing the relevant question, J. Forensic Sci. Soc., 24, 473–482, 1984.
91. Evett, I. W. and Lambert, J. A., The interpretation of refractive index measurements IV, Forensic Sci. Int., 24, 149–163, 1984.
92. Evett, I. W. and Lambert, J. A., The interpretation of refractive index measurements V, Forensic Sci. Int., 27, 97–110, 1985.
93. Robertson, B. and Vignaux, G. A., Probability — the logic of the law, Oxford
J. Legal Stud., 13, 457–478, 1993.
94. Robertson, B. and Vignaux, G. A., Interpreting Evidence: Evaluating Forensic
Science in the Courtroom, John Wiley & Sons, New York, 1995.
95. Finkelstein, M. O. and Fairley, W. B., A Bayesian approach to identification
evidence, Harv. Law Rev., 83, 489–517, 1970.
96. Aitken, C. G. G., Statistics and the Evaluation of Evidence for Forensic Scientists,
John Wiley & Sons, New York, 1995.
97. Evett, I. W., Bayesian inference and forensic science: problems and perspectives, Statistician, 36, 99–105, 1987.
98. Evett, I. W. and Buckleton, J. S., Some aspects of the Bayesian approach to
evidence evaluation, J. Forensic Sci. Soc., 29, 317–324, 1989.
99. Taroni, F., Champod, C., and Margot, P. A., Forerunners of Bayesianism in
early forensic science, Jurimetrics, 38, 183–200, 1998.
100. Bayes, T., An essay towards solving a problem in the doctrine of chances,
Philos. Trans. R. Soc. London for 1763, 53, 370–418, 1764.
101. Bernardo, J. M. and Smith, A. F. M., Bayesian Theory, John Wiley & Sons, New
York, 1994.
102. Evett, I. W. and Weir, B. S., Interpreting DNA Evidence, Sinauer Associates,
Sunderland, MA, 1998.
103. Evett, I. W., Lambert, J. A., and Buckleton, J. S., Further observations on glass
evidence interpretation, Sci. Justice, 35, 283–289, 1995.
©2000 CRC Press LLC
104. Lambert, J. A., Satterthwaite, M. J., and Harrison, P. H., A survey of glass
fragments recovered from clothing of persons suspected of involvement in
crime, Sci. Justice, 35, 273–281, 1995.
105. Harrison, P. H., Lambert, J. A., and Zoro, J. A., A survey of glass fragments
recovered from clothing of persons suspecrted of involvement in crime, Forensic Sci. Int., 27, 171–187, 1985.
106. Robertson, B. W. and Vignaux, G. A., DNA evidence: wrong answers or wrong
questions, Genetica, 96, 145–152, 1995.
107. McQuillan, J. and McCrossan, S., The Frequency of Occurrence of Glass Fragments in Head Hair Samples — a Pilot Investigation, Northern Ireland Forensic Science Laboratory, Belfast, Ireland, 1987.
108. McQuillan, J. and Edgar, K. A., Survey of the distribution of glass on clothing,
J. Forensic Sci. Soc., 32, 333–348, 1992.
109. Lindley, D. V., A statistical paradox, Biometrika, 44, 187–192, 1957.
110. Pearson, E. F., May, R. W., and Dabbs, M. D. G., Glass and paint fragments
found in men's outer clothing, J. Forensic Sci., 16, 283–300, 1971.
111. Davis, R. J. and DeHaan, J. D., A survey of men's footwear, J. Forensic Sci. Soc.,
17, 271–285, 1977.
112. Harrison, P. H. The Distribution of Refractive Index of Glass Fragments Found
on Shoes, Forensic Science Service, 1978.
113. Hoefler, K., Hermann, P., and Hansen, C. A Study of the Persistence of Glass
Fragments on Clothing After Breaking a Window, AN2FSS Symposium, Auckland, New Zealand, 1994.
114. Lau, L., Callowhill, B. C., Conners, N., Foster, K., Gorvers, R. J., Ohashi, K.
N., Sumner, A. M., and Wong, H., The frequency of occurrence of paint and
glass on the clothing of high school students, J. Can. Soc. Forensic Sci., 30,
233–240, 1997.
115. Zoro, J. A. and Fereday, M. J., Report of a Survey Concerning the Exposure
of Individuals to Breaking Glass, Forensic Science Service, Aldermaston, London, England, 1986.
116. Lambert, J. A. and Evett, I. W., The refractive index distribution of control
glass samples examined by the forensic science laboratories in the United
Kingdom, Forensic Sci. Int., 26, 1–23, 1984.
117. Faulkner, A. E. and Underhill, M., A Collection of Non-Sheet Glass from NonCasework Sources, Metropolitan Police Forensic Science Laboratory, London,
England, 1989.
118. Walsh, K. A. J. and Buckleton, J. S., On the problem of assessing the evidential
value of glass fragments embedded in footwear, J. Forensic Sci. Soc., 26, 55–60,
1986.
119. Lambert, J. A. and Satterthwaite, M. J., How likely is it that matching glass
will be found in these circumstances, FSS Report, 1–15, 1994.
120. Kirk, P. L., Crime Investigation, Interscience Publishers, New York, 1953.
121. Locke, J. and Unikowski, J. A., Breaking of flat glass, part 1: size and distribution of particles from plain glass windows, Forensic Sci. Int., 51, 251–262,
1991.
122. Locke, J. and Unikowski, J. A., Breaking of flat glass, part 2: effect of pane
parameters on particle distribution, Forensic Sci. Int., 56, 95–102, 1992.
123. Pounds, C. A. and Smalldon, K. W., The distribution of glass fragments in
front of a broken window and the transfer of fragments to individuals standing nearby, J. Forensic Sci. Soc., 18, 197–203, 1978.
©2000 CRC Press LLC
124. Luce, R. J. W., Buckle, J. L., and McInnis, I. A., Study on the backward fragmentation of window glass and the transfer of glass fragments to individual's
clothing, J. Can. Soc. Forensic Sci., 24, 79–89, 1991.
125. Zoro, J. A., Observations on the backward fragmentation of float glass, Forensic
Sci. Int., 22, 213–219, 1983.
126. Allen, T. J. and Scranage, J. K., The transfer of glass, part 1: transfer of glass
to individuals at different distances, Forensic Sci. Int., 93, 167–174, 1998.
127. Allen, T. J., Hoefler, K., and Rose, S. J., The transfer of glass, part 2: a study
of the transfer of glass to a person by various methods, Forensic Sci. Int., 93,
175–193, 1998.
128. Allen, T. J., Hoefler, K., and Rose, S. J., The transfer of glass, part 3: the transfer
of glass from a contaminated person to another uncontaminated person during a ride in a car, Forensic Sci. Int., 93, 195–200, 1998.
129. Allen, T. J., Hoefler, K., and Rose, S. J., The transfer of glass, part 4: the transfer
of glass fragments from the surface of an item to the person carrying it, Forensic
Sci. Int., 93, 201–208, 1998.
130. Francis, K., The Backward Fragmentation Pattern of Glass, as a Result of
Impact from a Firearm Projectile, Australian Federal Police, 1993.
131. Locke, J., Badmus, H. O., and Perryman, A. C., A study of the particles
generated when toughened windscreens disintegrate, Forensic Sci. Int., 31,
79–85, 1986.
132. Allen, T. J., Locke, J., and Scranage, J. K., Breaking of flat glass, part 4: size
and distribution of fragments from vehicle windscreens, Forensic Sci. Int., 93,
209–218, 1998.
133. Hicks, T., Vanina, R., and Margot, P., Transfer and persistence of glass fragments on garments, Sci. Justice, 36, 101–107, 1996.
134. Pounds, C. and Smalldon, K., The Distribution of Glass Fragments in Front
of a Broken Window and the Transfer of Fragments to Individuals Standing
Nearby, Forensic Science Service, 1977.
135. Underhill, M., The acquisition of breaking and broken glass, Sci. Justice, 37,
121–129, 1997.
136. Bone, R. G., Secondary Transfer of Glass Fragments Between Two Individuals,
Forensic Science Service, Birmingham, England, 1993.
137. Holcroft, G. A. and Shearer, B., The Transfer and Persistence of Glass Fragments with Special Reference to Secondary and Tertiary Transfer, Forensic
Science Service, Birmingham, England, 1993.
138. Brewster, F., Thorpe, J. W., Gettinby, G., and Caddy, B., The retention of glass
particles on woven fabrics, J. Forensic Sci., 30, 798–805, 1985.
139. Batten, R. A., Results of Window Breaking Experiments in the Birmingham
Laboratory, Forensic Science Service, Birmingham, England, 1989.
140. Cox, A. R., Allen, T. J., Barton, S., Messam, P., and Lambert, J. A., The Persistence of Glass. Part 1: the Effects of Cothing Type and Activity, CRSE, 1996.
141. Cox, A. R., Allen, T. J., Barton, S., Messam, P., and Lambert, J. A., The Persistence of Glass. Part 2: the Effects of Fragment Size, Forensic Science Service,
Birmingham, England, 1996.
142. Cox, A. R., Allen, T. J., Barton, S., Messam, P., and Lambert, J. A., The Persistence of Glass. Part 3: the Effects of Fragment Shape, Forensic Science Service,
Birmingham, England, 1996.
©2000 CRC Press LLC
143. Curran, J. M., Triggs, C. M., Hicks, T., Buckleton, J. S., and Walsh, K. A. J.,
Assessing transfer probabilities in a Bayesian interpretation of forensic glass
evidence, Sci. Justice, 38, 15–22, 1998.
144. Wright, S., The method of path coefficients, Ann. Math. Stat., 5, 161–215, 1934.
145. Spiegelhalter, D. J., Dawid, A. P., Lauritzen, S. L., and Cowell, R. G., Bayesian
analysis in expert systems, Stat. Sci., 8, 219–247, 1993.
146. Spiegelhalter, D. J., Thomas, A., Best, N., and Gilks, W., BUGS — Bayesian
Inference Using Gibbs Sampling, MRC Biostatistics Unit, Institute of Public
Health, Cambridge, U.K., 1994.
147. Sturges, H. A., The choice of a class interval, J. Am. Stat. Assoc., 21, 65–66, 1926.
148. Hoaglin, D. C., Mosteller, F., and Tukey, J. W., Understanding Robust and Exploratory Data Analysis, John Wiley & Sons, New York, 1983.
149. Dixon, W. J. and Kronmal, R. A., The choice of origin and scale for graphs, J.
Assoc. Comput. Mach., 12, 259–261, 1965.
150. Scott, D. W., On optimal and data-based histograms, Biometrika, 66, 605–610,
1979.
151. Freedman, D. and Diaconis, P., On the maximum deviation between the
histogram and and the underlying density, Z. Wahrscheinlichkeitstheorie verwandte Gebiete, 58, 139–167, 1981.
152. Freedman, D. and Diaconis, P., On the histogram as a density estimator: L2
theory, Z. Wahrscheinlichkeitstheorie verwandte Gebiete, 57, 453–456, 1981.
153. Scott, D. W., Multivariate Density Estimation, John Wiley & Sons, New York,
1992.
154. Curran, J. M., Triggs, C. M., Almirall, J. R., Buckleton, J. S., and Walsh, K. A.
J., The interpretation of elemental composition measurements from forensic
glass evidence, Sci. Justice, 37, 241–244, 1997.
155. Curran, J. M., Triggs, C. M., Almirall, J. R., Buckleton, J. S., and Walsh, K. A.
J., The interpretation of elemental composition measurements from forensic
glass evidence II, Sci. Justice, 37, 245–249, 1997.
156. Buckleton, J. S. and Walsh, K. A. J., The Use of statistics in Forensic Science,
Aitken, C. G. G. and Stoney, D. A., Eds., Ellis Horwood Ltd., Chichester, 1991.
157. Taroni, F. and Aitken, C. G. G., Forensic science at trial, Jurimetrics, 37, 327–337,
1997.
158. Evett, I. W., Towards a uniform framework for reporting opinions in forensic
science casework, Sci. Justice, 38, 198–202, 1998.
159. Efron, B., Bootstrap methods: another look at the jackknife, Ann. Stat., 7, 1–26,
1979.
160. Thompson, W. C. and Schumann, E. L., Interpretation of statistical evidence
in criminal trials: the prosecutor's fallacy and the defence attorney's fallacy,
Law Hum. Behav., 11, 167–187, 1987.
161. Aitken, C. G. G. and Stoney, D. A., The Use of Statistics in Forensic Science, Ellis
Horwood Ltd., Chichester, 1991.
162. Evett, I. W., Avoiding the transposed conditional, Sci. Justice, 35, 127–131, 1995.
©2000 CRC Press LLC